Chapter 5 - Functions
This chapter discusses functions. It contains the following sections:- section 5.1 - In this section we introduce functions,
the function machine concept, functional notation and different ways to express functions.
We also talk about composite functions and inverse functions.
- section 5.2 - This is a reference section that
shows the graphs of eight types of functions.
- section 5.3 - This is a reference section that gives many details about the eleven functions that are built into the Algebra Coach and into all scientific calculators.
5.1 - Introduction to functions
- Definition of function
- The function machine concept and functional notation
- Ways of expressing a function: list form, graph form, formula form
- The argument and value of a function
- Identifying the domain and range of a function
- The vertical line test for a function
- One-to-one and many-to-one functions
- Substituting expressions into functions
- Composition of functions
- Inverse of a function
Definition: A function is a correspondence or mapping from a first set of numbers, called the domain of the function, to a second set of numbers, called the range of the function, such that for each member of the domain there is exactly one member of the range, as shown in this picture: |
The function machine concept and functional notation
It is useful to think of a function as a machine with a number from the domain as the input and a corresponding number of the range as output. The function is given a name like f (short for the word “function”), and if the number going into the machine is called x, then the corresponding number returned by or coming out of the machine is denoted f (x). Here is a picture:The functional notation f (x) literally means “function of x”.
Ways of expressing a function
A function can be expressed in various ways:-
A function can be expressed in list form, especially if the
domain and range are small. Here is an example of a function in list form:
domain range 10 3 2 15 5 46
- A function can be expressed in graph form.
The function is represented by a curve drawn on a cartesian plane.
The domain is plotted horizontally (in the x direction) and the range is
plotted vertically (in the y direction).
To find the range value y corresponding to a given domain value x
you start at the domain value on the x axis, go vertically
until you reach the graph, then go horizontally until you reach the
y axis. Here is an example of a function in graph form:
If this function is called g then for example g (−1) = 1 and g (2) = 2.5. Click here to see the graphs of a variety of function types.
- A function can be expressed in formula form.
The formula is used to calculate the range value for any given domain value.
Here is an example of a function in formula form:
h (x) = x^{ 2} − 2 x
This function is called h . Here is an example showing how the formula is used to calculate a value of the range for a value of the domain, say 4. The domain value 4 is substituted in for x wherever x occurs and then the formula is simplified to yield the range value:h (4) = 4^{ 2} − 2 · 4 = 8.
Here is another example with the domain value 5:h (5) = 5^{ 2} − 2 · 5 = 15.
Another way to write the above function is this:y = x^{ 2} − 2 x.
In this form h (x) has been replaced by a new variable y so that there are now two variables, x and y. Variable y is the value of the range that corresponds to the value of variable x of the domain. Variable y is called the dependent variable and variable x is called the independent variable. This form plays down the function aspect of the relationship and just gives an equation connecting values of the domain and range. Yet another way to write the function is in two parts, like this:y = h (x), where h (x) = x^{ 2} − 2 x.
The first part gives a name to the function and the second part gives the formula for the function.
The argument and value of a function
The value of the domain that goes into the function machine is also called the argument of the function and the value of the range that comes out of the function machine is also called the value of the function. For example suppose that f (5) = 15. Then we say that the argument of the function f is 5 and the value of f is 15.Identifying the domain and range of a function
The domain and range of a function isn’t always the set of all real numbers. If a function is expressed in list or graph form you can identify the domain and range by simply looking at the list or graph. But if the function is expressed in formula form then you must do the following:- Substitute potential domain values into the formula and make sure
that they don’t cause an undefined operation to occur (such as division
by zero or the square root of a negative number). If they do then they are
not in the domain.
- Once the domain is known, you can find the range by substituting various domain value into the formula.
Example: Consider the function . The domain must be because otherwise we are trying to take the square root of a negative number. Then if we substitute various values of the domain into the formula, we see that the range is . Here is a graph of this function which corroborates our findings:
The vertical line test for a function
The definition of function states that for each member of the domain there can be only one member of the range. Thus the graph of a function cannot look like this:where there is an x value for which there are two or more corresponding y values. If the graph does not pass this so-called vertical line test then it is not the graph of a function. Instead we say that it is the graph of a relation between x and y.
One-to-one and many-to-one functions
A function is said to be one-to-one if every y value has exactly one x value mapped onto it, and many-to-one if there are y values that have more than one x value mapped onto them. This graph shows a many-to-one function. The three dots indicate three x values that are all mapped onto the same y value.One complication with a many-to-one function is that it can’t have an inverse function. If it could, that inverse would be one-to-many and this would violate the definition of a function.
Substituting expressions into functions
Often, especially in calculus, we use the formula form of a function and we let the argument be an expression instead of just a number. The only complication in this case is that we must usually put brackets around the argument to preserve the proper order of operations. This is because the formula is just a recipe for what to do to the input (the argument) to get the output (the function value). For example the functional notation:f (x) = x^{ 2} − 2 xmeans that the function value is gotten by taking the square of the argument and subtracting twice the argument from it. It doesn’t actually matter what letter we use for the argument; it is how the function works that is important.
Thus the following are all valid substitutions:
- f (4) = 4^{ 2} − 2 · 4
- f (t) = t^{ 2} − 2 t
- f (x+ h) = (x+ h)^{ 2} − 2 (x+ h)
- f (a x) = (a x)^{ 2} − 2 (a x)
Warning: Don’t be confused by the brackets. On the left side of each example the brackets indicate functional notation. Thus:
f (whatever)means that we have a function named f and that its argument is whatever. We are not multiplying f by whatever ! On the right side we are using brackets to preserve the order of operations.
Composition of functions
Just as we can substitute an expression into a function, so we can substitute another function into a function. For example in the previous section we defined the function:f (x) = x^{ 2} − 2 xIf we substitute another function g (x) into this function then we get: For example if g (x) = x + 3 then: We can also switch the order and substitute f into g, like this: Notice that the result is completely different. If we think of f and g as machines, then substituting f into g means that the output of f is the input of g, as shown here: The composition of functions is important because this method can be used to create complicated functions out of simple components.
Inverse of a function
Suppose that a function f maps x onto y and that another function g maps y back onto the original x as shown here: Then function g is called the inverse function of function f and the composition of f and g has no overall effect. Note that function f must be one-to-one for it to have an inverse.One way to derive the inverse function g for any function f is this:
- Set f (x) equal to y.
- Solve the equation y = f (x) for x.
If there is exactly one solution then the inverse exists; otherwise it doesn’t.
- In the equation just found, rename x to be g (y).
Example: Find the inverse function g of the function f (x) = 2 x + 3.
Example: Try to find the inverse function of the function f (x) = x^{ 2}.Notice that function f takes its argument, multiplies it by 2 and then adds 3. The inverse function, g, does exactly the opposite steps in the opposite order. It takes its argument, first subtracts 3 and then divides by 2. This is exactly what you would expect the inverse to do.
Set f (x) equal to y
Solve for x
Rename x as g (y). This is the inverse.
Notice that f maps two points onto every point. For example f (2) = 4 and f (−2) = 4. Thus the inverse would have to map the point 4 back to both points 2 and −2. But this violates the definition of a function so there is no inverse.
Set f (x) equal to y
Solve for x. There are two solutions so the inverse doesn’t exist.
5.2 - Reference - Graphs of eight basic types of functions
The purpose of this reference section is to show you graphs of various types of functions in order that you can become familiar with the types. You will discover that each type has its own distinctive graph. By showing several graphs on one plot you will be able to see their common features. Examples of the following types of functions are shown in this gallery: In each case the argument (input) of the function is called x and the value (output) of the function is called y.Linear functions. These are functions of the form:
y = m x + b,where m and b are constants. A typical use for linear functions is converting from one quantity or set of units to another. Graphs of these functions are straight lines. m is the slope and b is the y intercept. If m is positive then the line rises to the right and if m is negative then the line falls to the right. Linear functions are described in detail here.
Quadratic functions. These are functions of the form:
y = a x^{ 2} + b x + c,where a, b and c are constants. Their graphs are called parabolas. This is the next simplest type of function after the linear function. Falling objects move along parabolic paths. If a is a positive number then the parabola opens upward and if a is a negative number then the parabola opens downward. Quadratic functions are described in detail here.
Power functions. These are functions of the form:
y = a x^{ b},where a and b are constants. They get their name from the fact that the variable x is raised to some power. Many physical laws (e.g. the gravitational force as a function of distance between two objects, or the bending of a beam as a function of the load on it) are in the form of power functions. We will assume that a = 1 and look at several cases for b:
The power b is a positive integer. See the graph to the right. When x = 0 these functions are all zero. When x is big and positive they are all big and positive. When x is big and negative then the ones with even powers are big and positive while the ones with odd powers are big and negative.
The power b is a negative integer. See the graph to the right. When x = 0 these functions suffer a division by zero and therefore are all infinite. When x is big and positive they are small and positive. When x is big and negative then the ones with even powers are small and positive while the ones with odd powers are small and negative.
The power b is a fraction between 0 and 1. See the graph to the right. When x = 0 these functions are all zero. The curves are vertical at the origin and as x increases they increase but curve toward the x axis.
The power function is discussed in detail here.
Polynomial functions. These are functions of the form:
y = a_{n} · x^{ n} + a_{n −1} · x^{ n −1} + … + a_{2} · x^{ 2} + a_{1} · x + a_{0},where a_{n}, a_{n −1}, … , a_{2}, a_{1}, a_{0} are constants. Only whole number powers of x are allowed. The highest power of x that occurs is called the degree of the polynomial. The graph shows examples of degree 4 and degree 5 polynomials. The degree gives the maximum number of “ups and downs” that the polynomial can have and also the maximum number of crossings of the x axis that it can have.
Polynomials are useful for generating smooth curves in computer graphics applications and for approximating other types of functions. Polynomials are described in detail here.
Rational functions. These functions are the ratio of two polynomials. One field of study where they are important is in stability analysis of mechanical and electrical systems (which uses Laplace transforms).
When the polynomial in the denominator is zero then the rational function becomes infinite as indicated by a vertical dotted line (called an asymptote) in its graph. For the example to the right this happens when x = −2 and when x = 7.
When x becomes very large the curve may level off. The curve to the right levels off at y = 5.
The graph to the right shows another example of a rational function. This one has a division by zero at x = 0. It doesn't level off but does approach the straight line y = x when x is large, as indicated by the dotted line (another asymptote).
Exponential functions. These are functions of the form:
y = a b^{ x},where x is in an exponent (not in the base as was the case for power functions) and a and b are constants. (Note that only b is raised to the power x; not a.) If the base b is greater than 1 then the result is exponential growth. Many physical quantities grow exponentially (e.g. animal populations and cash in an interest-bearing account).
If the base b is smaller than 1 then the result is exponential decay. Many quantities decay exponentially (e.g. the sunlight reaching a given depth of the ocean and the speed of an object slowing down due to friction).
Exponential functions are described in detail here.
Logarithmic functions. There are many equivalent ways to define logarithmic functions. We will define them to be of the form:
y = a ln (x) + b,where x is in the natural logarithm and a and b are constants. They are only defined for positive x. For small x they are negative and for large x they are positive but stay small. Logarithmic functions accurately describe the response of the human ear to sounds of varying loudness and the response of the human eye to light of varying brightness. Logarithmic functions are described in detail here.
Sinusoidal functions. These are functions of the form:
y = a sin (b x + c),where a, b and c are constants. Sinusoidal functions are useful for describing anything that has a wave shape with respect to position or time. Examples are waves on the water, the height of the tide during the course of the day and alternating current in electricity. Parameter a (called the amplitude) affects the height of the wave, b (the angular velocity) affects the width of the wave and c (the phase angle) shifts the wave left or right. Sinusoidal functions are described in detail here.
5.3 - Reference - The elementary functions
Mathematicians have defined literally hundreds of functions. (See the Handbook of Mathematical Functions, by Abramowitz and Stegun, for definitions of many of them.) The eleven most useful functions are built into the Algebra Coach. They are:
sine
(sin)cosine
(cos)tangent
(tan)arcsine
(arcsin)arccosine
(arccos)arctangent
(arctan)base 10 logarithm
(log)natural logarithm
(ln)exponential with base e
(exp or e^{ x} )square root
(sqrt)power
( y^{ x} )
For each of these functions we will:
- explain its significance
- state the domain and range of the function
- show a graph of the function
- list special values of the function
- give an algorithm to calculate the function
- show how to extend the function to the complex numbers
- explain how to use the function in the Algebra Coach
The sine, cosine and tangent functions
Background: In what follows we assume that you are familiar with trigonometry. The sine, cosine and tangent functions (denoted sin, cos and tan) are important in trigonometry and many other areas of mathematics. Here is how they are derived. Consider the vector (the red arrow) in the picture to the right. It has its tail at the origin, has length r and is oriented at angle θ.Let (x, y) denote the coordinates of the head of the vector (i.e. let x and y be the movements in the x and then in the y direction required to get from the tail to the head of the vector.) The three arrows form a triangle in standard position.
Now imagine changing the angle θ. The vector will point in another direction but its head will still be somewhere on the dotted circle (because its length r is unchanged).
The values of x and y will change. For example in the picture to the right the values of x and y are both negative.
Definitions: The sine, cosine and tangent functions (denoted sin, cos and tan) are defined as returning the following ratios: These ratios are functions of θ because x and y change with θ. |
Graph of the sine function: The picture on the left shows the red vector pointing at various angles θ and the graph on the right shows the resulting function sin (θ):
Graph of the cosine function: The next picture on the left again shows the red vector pointing at various angles θ and the graph on the right shows the resulting function cos (θ):
Graph of the tangent function: The next graph shows the function tan (θ). The dotted vertical lines are asymptotes (lines that the function approaches but never touches):
In the above three graphs the angle θ is measured in radians. If you want θ to be measured in degrees then simply change the horiontal scale so that θ runs from 0 to 360° instead of from 0 to 2π radians; the shapes of the graphs are otherwise unchanged. The sine, cosine and tangent functions are said to be periodic. This means that they repeat themselves in the horizontal direction after a certain interval called a period. The sine and cosine functions have a period of 2π radians and the tangent function has a period of π radians.
Domain and range: From the graphs above we see that for both the sine and cosine functions the domain is all real numbers and the range is all reals from −1 to +1 inclusive. For the tangent function the domain is all real numbers except ±π/2, ±3π/2, ±5π/2, …, (or in degrees: ±90°, ±270°, ±450°, …), where the tangent function is undefined. The range of the tangent function is all real numbers.
The definitions of sine, cosine and tangent can be extended to the complex numbers by defining the functions by their Taylor series instead of by the ratio of two lengths. In that case, the domain and range of the sine and cosine functions is all complex numbers, and the domain of the tangent function is all complex numbers except ±π/2, ±3π/2, ±5π/2, …, where the tangent function is undefined, and the range is all complex numbers.
Special values: For the two triangles shown below, Pythagoras’ theorem gives simple, exact values for the lengths of the sides and hence for the values of the sine, cosine and tangent functions. The following table gives these values as well as those for angles of 0° and 90° :
Algorithms for calculating sine, cosine and tangent: Have you ever wondered how calculators and computers are able to calculate functions like sine, cosine and tangent? The answer is that they make use of formulas like these:
These formulas are called polynomial approximations and are based on Taylor's series. To use them x must be in radians. They are very accurate when x is close to 0 but lose accuracy as x gets bigger. When x = π/4 radians (i.e. 45°) the sin formula is only accurate to within ±0.00004, cos to within ±0.000004 and tan to within ±0.004.
If x is greater than π/4 these formulas are too inaccurate to be used directly. Instead cofunctions and symmetries of the sine, cosine and tangent functions are exploited to reduce the angle x and improve the accuracy. For example, to calculate sin(440°), use is made of the fact that this is the same as sin(80°), which is the same as cos(10°) which is the same as cos(0.174533 radians), which is then computed using the cos formula. Click here to see algorithms that computers use for calculating the sine function, the cosine function and the tangent function.
How to use the sine, cosine and tangent functions in the Algebra Coach
- Type sin(x), cos(x) or tan(x) into the textbox, where x is the argument.
The argument must be enclosed in brackets.
- Set the relevant options:
- Set the exact / floating point option. (Exact mode lets you use special values.)
- Set the degree / radian mode option. (Radian mode is more versatile and recommended. Then any angles that you enter are assumed to be in radians, but you can still enter angles in degrees by following them with the letter d; see the next bullet.)
- Set the d does / does not represent the ° symbol option. (This option is only available in radian mode. When this option is on you can, for example, type in cos(30d+2) to mean the cosine of 30 degrees plus 2 radians.)
- Set the p does / does not represent π option.
- Turn on complex numbers if you want to be able to evaluate
the sine, cosine or tangent of a complex number.
- Click the Simplify button.
The arcsine function
Background: The arcsine function is the inverse of the sine function (as long as the sine function is restricted to a certain domain). Click here for a review of inverse functions.Recall that the sine function takes an angle x as input and returns the sine of that angle as output: For example if 30° is the input then 0.5 is the output. Here we want to create the inverse function that would take 0.5 as input and return 30° as output. But there is a problem. Notice that there are many angles whose sine is 0.5: We say that this mapping is many-to-one. This means that the inverse mapping would be one-to-many and therefore would not satisfy the ‘one range value’ requirement for a mapping to be a function. To fix this problem we introduce both a relation called Arcsine (with capital A and abbreviation Arcsin) and a function called arcsine (with lower case ‘a’ and abbreviation arcsin). Here is how they are defined:
Definition: The Arcsine of x, denoted Arcsin(x), is defined as ‘the set of all angles whose sine is x’. It is a one-to-many relation. Here is an example: Definition: The arcsine of x, denoted arcsin(x), is defined as ‘the one angle between −π/2 and +π/2 radians (or between −90° and +90°) whose sine is x’. It is a one-to-one function. Here is an example: Definition: Of all the values returned by the Arcsine relation, the one that is the same as the value returned by the arcsine function is called the principal value of the Arcsine relation. (An example is the value 30° shown above in red.)
Graph: The red curve in the graph to the right is the arcsine function. Notice that for any x between −1 and +1 it returns a single value between −π/2 and +π/2 radians.
If we add the gray curve to the red curve then we get a graph of the Arcsine relation. A vertical line drawn anywhere between x = −1 and +1 would touch this curve at many places and this means that the Arcsine relation would return many values.
In this graph the angle y is measured in radians. If you want to measure y in degrees then simply change the vertical scale so that y runs from −360° to 360° instead of from −2π to 2π radians; the shape of the graph is otherwise unchanged.
If you compare the Arcsine graph to the sine graph then you see that one can be gotten from the other by interchanging the horizontal and vertical axes.
Domain and range: The domain of the arcsine function is from −1 to +1 inclusive and the range is from −π/2 to π/2 radians inclusive (or from −90° to 90°).
The arcsine function can be extended to the complex numbers, in which case the domain is all complex numbers.
Special values of the arcsine function (Click here for more details)
Solving the Equation sin(θ) = c for θ by using arcsine and Arcsine
Suppose that an angle θ is unknown but that its sine is known to be c. Then finding that angle requires solving this equation for θ:
sin (θ) = cIf this is, say, a simple right triangle problem and we know that the angle θ must be somewhere between 0 and 90° then the solution is the single value:
θ = arcsin (c)On the other hand, if this is a more advanced problem and we need to find all the possible angles whose sine is c then the solution is the entire set of values:
θ = Arcsin (c)The solutions in these two cases follow directly from the definitions of the arcsine function and Arcsine relation. Note that if c is greater than 1 or less than −1 then there are no real solutions. However there are complex solutions.
Evaluating Arcsin(c)
If c is a number then the entire set of values of Arcsin(c) can be found using the following procedure. Refer to the graph to the right where the dots are the desired values.
- The first value (the principal value), denoted θ_{PV} , is found by
evaluating arcsin(c) with a calculator or with the Algebra Coach.
- The second value, called θ_{2} , is found by
using the symmetry of the Arcsine curve. Notice that the two blue arrows in the
graph have the same length. This means that θ_{2} is just as far below π as
θ_{PV} is above zero. In formula form:
θ_{2} = π − θ_{PV}
(Click here to see the CAST method for finding θ_{2} .) - All the other values above and below these two values
can be found from these two values by adding or
subtracting multiples of 2π. If we use the integer n to count
which multiple then the other values can be gotten from this formula:
For example if we let n = −1 then we get values for
the two lowest dots in the graph.
- If you are using degrees instead of radians then use the following formulas instead of the previous ones:
How to use the arcsine function in the Algebra Coach
- Type arcsin(x) into the textbox, where x is the argument.
The argument must be enclosed in brackets.
- Set the relevant options:
- Set the arcsin, arccos and arctan option. (The return principal value setting returns one value; the don't evaluate setting is useful if you want all the values of the Arc relation - but you will have to calculate them yourself.)
- Set the exact / floating point option. (Exact mode lets you use special values.)
- Set the degree / radian mode option.
- Set the p does / does not represent π option. (If you want arcsine to return special values in radian mode then turn this on.)
- Turn on complex numbers if you want to be able to evaluate
the arcsine of a complex number or of a number bigger than 1.
- Click the Simplify button.
Algorithm for the arcsine function
Click here to see the algorithm that computers use to evaluate the arcsine function.
The arccosine function
Background: The arccosine function is the inverse of the cosine function (as long as the cosine function is restricted to a certain domain). Click here for a review of inverse functions.Recall that the cosine function takes an angle x as input and returns the cosine of that angle as output: For example if 60° is the input then 0.5 is the output. Here we want to create the inverse function that would take 0.5 as input and return 60° as output. But there is a problem. Notice that there are many angles whose cosine is 0.5: We say that this mapping is many-to-one. This means that the inverse mapping would be one-to-many and therefore would not satisfy the ‘one range value’ requirement for a mapping to be a function. To fix this problem we introduce both a relation called Arccosine (with capital A and abbreviation Arccos) and a function called arccosine (with lower case ‘a’ and abbreviation arccos). Here is how they are defined:
Definition: The Arccosine of x, denoted Arccos(x), is defined as ‘the set of all angles whose cosine is x’. It is a one-to-many relation. Here is an example: Definition: The arccosine of x, denoted arccos(x), is defined as ‘the one angle between 0 and π radians (or between 0° and 180°) whose cosine is x’. It is a one-to-one function. Here is an example: Definition: Of all the values returned by the Arccosine relation, the one that is the same as the value returned by the arccosine function is called the principal value of the Arccosine relation. (An example is the value 60° shown above in red.)
Graph: The red curve in the graph to the right is the arccosine function. Notice that for any x between −1 and +1 it returns a single value between 0 and +π radians.
If we add the gray curve to the red curve then we get a graph of the Arccosine relation. A vertical line drawn anywhere between x = −1 and +1 would touch this curve at many places and this means that the Arccosine relation would return many values.
In this graph the angle y is measured in radians. If you want to measure y in degrees then simply change the vertical scale so that y runs from −360° to 360° instead of from −2π to 2π radians; the shape of the graph is otherwise unchanged.
If you compare the Arccosine graph to the cosine graph then you see that one can be gotten from the other by interchanging the horizontal and vertical axes.
Domain and range: The domain of the arccosine function is from −1 to +1 inclusive and the range is from 0 to π radians inclusive (or from 0° to 180°).
The arccosine function can be extended to the complex numbers, in which case the domain is all complex numbers.
Special values of the arccosine function (Click here for more details)
Solving the Equation cos(θ) = c for θ by using arccosine and Arccosine
Suppose that an angle θ is unknown but that its cosine is known to be c. Then finding that angle requires solving this equation for θ:
cos (θ) = cIf this is, say, a simple right triangle problem and we know that the angle θ must be somewhere between 0 and 90° then the solution is the single value:
θ = arccos (c)On the other hand, if this is a more advanced problem and we need to find all the possible angles whose cosine is c then the solution is the entire set of values:
θ = Arccos (c)The solutions in these two cases follow directly from the definitions of the arccosine function and Arccosine relation. Note that if c is greater than 1 or less than −1 then there are no real solutions. However there are complex solutions.
Evaluating Arccos(c)
If c is a number then the entire set of values of Arccos(c) can be found using the following procedure. Refer to the graph to the right where the dots are the desired values.
- The first value (the principal value), denoted θ_{PV} , is found by
evaluating arccos(c) with a calculator or with the Algebra Coach.
- The second value, called θ_{2} , is found by
using the symmetry of the Arccosine curve. Notice that the two blue arrows in the
graph have the same length. This means that θ_{2} is just as far below 2π as
θ_{PV} is above zero. In formula form:
θ_{2} = 2 π − θ_{PV}
(Click here to see the CAST method for finding θ_{2} .) - All the other values above and below these two values
can be found from these two values by adding or
subtracting multiples of 2π. If we use the integer n to count
which multiple then the other values can be gotten from this formula:
For example if we let n = −1 then we get values for
the two lowest dots in the graph.
- If you are using degrees instead of radians then use the following formulas instead of the previous ones:
How to use the arccosine function in the Algebra Coach
- Type arccos(x) into the textbox, where x is the argument.
The argument must be enclosed in brackets.
- Set the relevant options:
- Set the arcsin, arccos and arctan option. (The return principal value setting returns one value; the don't evaluate setting is useful if you want all the values of the Arc relation - but you will have to calculate them yourself.)
- Set the exact / floating point option. (Exact mode lets you use special values.)
- Set the degree / radian mode option.
- Set the p does / does not represent π option. (If you want arccosine to return special values in radian mode then turn this on.)
- Turn on complex numbers if you want to be able to evaluate
the arccosine of a complex number or of a number bigger than 1.
- Click the Simplify button.
Algorithm for the arccosine function
Click here to see the algorithm that computers use to evaluate the arccosine function.
The arctangent function
Background: The arctangent function is the inverse of the tangent function (as long as the tangent function is restricted to a certain domain). Click here for a review of inverse functions.Recall that the tangent function takes an angle x as input and returns the tangent of that angle as output: For example if 45° is the input then 1.0 is the output. Here we want to create the inverse function that would take 1.0 as input and return 45° as output. But there is a problem. Notice that there are many angles whose tangent is 1.0: We say that this mapping is many-to-one. This means that the inverse mapping would be one-to-many and therefore would not satisfy the ‘one range value’ requirement for a mapping to be a function. To fix this problem we introduce both a relation called Arctangent (with capital A and abbreviation Arctan) and a function called arctangent (with lower case ‘a’ and abbreviation arctan). Here is how they are defined:
Definition: The Arctangent of x, denoted Arctan(x), is defined as ‘the set of all angles whose tangent is x’. It is a one-to-many relation. Here is an example: Definition: The arctangent of x, denoted arctan(x), is defined as ‘the one angle between −π/2 and +π/2 radians (or between −90° and +90°) whose tangent is x’. It is a one-to-one function. Here is an example: Definition: Of all the values returned by the Arctangent relation, the one that is the same as the value returned by the arctangent function is called the principal value of the Arctangent relation. (An example is the value 45°shown above in red.)
Graph: The red curve in the graph to the right is the arctangent function. Notice that for any x it returns a single value between −π/2 and +π/2 radians.
If we add the gray curves to the red curve then we get a graph of the Arctangent relation. A vertical line drawn anywhere would touch this set of curves at many places and this means that the Arctangent relation would return many values.
In this graph the angle y is measured in radians. If you want to measure y in degrees then simply change the vertical scale so that y runs from −180° to 180° instead of from −π to π radians; the shape of the graph is otherwise unchanged.
If you compare the Arctangent graph to the tangent graph then you see that one can be gotten from the other by interchanging the horizontal and vertical axes.
Domain and range: The domain of the arctangent function is all real numbers and the range is from −π/2 to π/2 radians exclusive (or from −90° to 90°).
The arctangent function can be extended to the complex numbers, in which case the domain is all complex numbers.
Special values of the arctangent function (Click here for more details)
Solving the Equation tan(θ) = c for θ by using arctangent and Arctangent
Suppose that an angle θ is unknown but that its tangent is known to be c. Then finding that angle requires solving this equation for θ:
tan (θ) = cIf this is, say, a simple right triangle problem and we know that the angle θ must be somewhere between 0 and 90° then the solution is the single value:
θ = arctan (c)On the other hand, if this is a more advanced problem and we need to find all the possible angles whose tangent is c then the solution is the entire set of values:
θ = Arctan (c)The solutions in these two cases follow directly from the definitions of the arctangent function and Arctangent relation.
Evaluating Arctan(c)
If c is a number then the entire set of values of Arctan(c) can be found using the following procedure. Refer to the graph to the right where the dots are the desired values.
- The first value (the principal value), denoted θ_{PV} , is found by
evaluating arctan(c) with a calculator or with the Algebra Coach.
- All the other values above and below this value can be found by
using the fact that adjacent values are separated from each other by a distance of π.
If we use the integer n to count multiples of π then the other
values can be gotten from this formula:
θ = θ_{PV} + π n
(Click here to see the CAST method for finding θ_{2} .) - If you are using degrees instead of radians then use the following formulas
instead of the previous ones:
θ = θ_{PV} + 180° · n
How to use the arctangent function in the Algebra Coach
- Type arctan(x) into the textbox, where x is the argument.
The argument must be enclosed in brackets.
- Set the relevant options:
- Set the arcsin, arccos and arctan option. (The return principal value setting returns one value; the don't evaluate setting is useful if you want all the values of the Arc relation - but you will have to calculate them yourself.)
- Set the exact / floating point option. (Exact mode lets you use special values.)
- Set the degree / radian mode option.
- Set the p does / does not represent π option. (If you want arctangent to return special values in radian mode then turn this on.)
- Turn on complex numbers if you want to be able to evaluate
the arctangent of a complex number.
- Click the Simplify button.
Algorithm for the arctangent function
Click here to see the algorithm that computers use to evaluate the arctangent function.
The base 10 logarithm function
Background: Every positive number, y, can be expressed as 10 raised to some power, x. This relationship is described by the equationy = 10^{ x},and described by this graph: For example the number 16 can be expressed as 10^{ 1.2}. This is the black dot in the graph. We define a function called the base 10 logarithm that takes a number like 16 as input, calculates that it can be expressed as 10^{ 1.2}, and returns the exponent 1.2 as its output value: Here is the formal definition of the base 10 logarithm function.
Definition: The base 10 logarithm is the function that takes any positive number x as input and returns the exponent to which the base 10 must be raised to obtain x. It is denoted log(x). |
Example 1: Evaluate log ( 1000 ).
The argument of the logarithm function (i.e. the quantity in brackets, 1000), is easily expressed as 10 raised to the exponent 3. The logarithm function then returns the exponent.
log ( 1000 ) = log ( 10^{ 3} ) = 3
Example 2: Evaluate log ( 10^{ 5.7} ).
The argument is already expressed as 10 raised to an exponent, so the logarithm function simply returns the exponent.
log ( 10^{ 5.7} ) = 5.7
Example 3: Evaluate log ( 16 ).
According to the graph 16 = 10^{ 1.2}. The logarithm function returns the exponent 1.2.
log ( 16 ) = log ( 10^{ 1.2} ) = 1.2
Graph: The blue curve shown to the right is the graph of the base 10 logarithm function, y = log(x). Notice that for any positive x it is single valued and for any negative x it is undefined. If you compare this graph to the graph of y = 10^{ x} above then you see that one can be gotten from the other by interchanging the x and y axes.
For comparison the red curve is the graph of the natural logarithm function (y = ln(x), covered in the next section). The natural logarithm graph has exactly the same shape as the base 10 logarithm graph; it is just 2.3 times as tall.
An important feature of logarithm functions (no matter what base) is that they increase very slowly as x becomes very large. They describe nicely how the human ear perceives loudness and the way the human eye perceives brightness.
Domain and range: The domain of the base 10 logarithm function is all positive real numbers and the range is all real numbers.
The base 10 logarithm function can be extended to the complex numbers, in which case the domain is all complex numbers except zero. The base 10 logarithm of zero is always undefined.
Some special values of the base 10 logarithm function
Solving the equation 10^{ x} = c for x by using the base 10 logarithm function
Suppose that x is unknown but that 10^{ x} equals a known value c. Then finding x requires solving the following equation for x.
10^{ x} = cThe solution is
x = log (c)This is because finding log (c) means expressing c as 10 to an exponent, and then returning that exponent. But the original equation says that that exponent is x. Note that if c is negative then there is no real solution. However there is a complex solution. Furthermore if c = 0 then there is no solution at all.
log (x) and 10^{ x} are inverse functions
Consider the 10^{ x} function which takes x and returns 10^{ x}, like this: The base 10 logarithm function is defined to do exactly the opposite, namely: Therefore these are inverse functions.
Note the following:
- Because the 10^{ x} function is the inverse of
the base 10 logarithm function it is sometimes called the antilogarithm function.
- We saw above that the solution of
10^{ x} = y is
x = log (y). We should look at these two equations
as expressing the same relationship between x and y but
from different points of view. The first equation is the relationship solved for
y and the second one is the relationship solved for x.
(An analogy is that the statement “Tom is Jane’s brother”
is equivalent to the statement that “Jane is Tom’s sister”.)
- In the previous bullet we saw that the two equations,
10^{ x} = y and
x = log (y), said the same thing.
If we replace x in the first equation
by the x of the second equation we get this identity:
10^{ log( y)} = y
and if we replace y in the second equation by the y of the first equation we get this identity:x = log (10^{ x })
These identities are useful for showing how the logarithm and antilogarithm cancel each other.
- If you compare the graph
of y = log (x) to the
graph
of y = 10^{ x}
then you see that one can be gotten from the other by interchanging
the x and y axes. This always happens with inverse functions.
How to use the base 10 logarithm function in the Algebra Coach
- Type log(x) into the textbox, where x is the argument.
The argument must be enclosed in brackets.
- Set the relevant options:
- Set the exact / floating point option. In floating point mode the base 10 logarithm of any number is evaluated. In exact mode the base 10 logarithm of an integer is not evaluated because doing so would result in an approximate number.
- Turn on complex numbers if you want to be able to evaluate the
base 10 logarithm of a negative or complex number.
- Click the Simplify button.
Algorithm for the base 10 logarithm function
Click here to see the algorithm that computers use to evaluate the base 10 logarithm function.
The natural logarithm function
Background: You might find it useful to read the previous section on the base 10 logarithm function before reading this section. The two sections closely parallel each other.Recall that the base 10 logarithm function takes a number like 16 as input, calculates that it can be written as 10^{ 1.2}, and then returns the exponent 1.2 as its output value. But why use base 10? After all, probably the only reason that the number 10 is important to humans is that they have 10 fingers with which they first learned to count. Maybe on some other planet populated by 8-fingered beings they use base 8!
In fact probably the most important number in all of mathematics (click here to see why) is the number 2.71828…, which we give the name e, in honor of Leonard Euler, who first discovered it. It will be important to be able to take any positive number, y, and express it as e raised to some power, x. We can write this relationship in equation form:
y = e^{ x}For example 5 can be written as e^{ 1.6} (the exponent is approximate). How do we know that this is the correct power of e? Because we get it from the graph shown below.
To make this graph we made a table of a few obvious values of y = e^{ x} as shown below, left. Then we plotted the values in the graph (they are the red dots) and drew a smooth curve through them. Then we observed that the curve went through y = 5 and x = 1.6 (the black dot). This means that 5 = e^{ 1.6}.
If you compare this graph to that of y = 10^{ x} you see that both have the same so-called exponential growth shape but that this graph grows more slowly.
We next define a function called the natural logarithm that takes a number like 5 as input, calculates that it can be written as e^{ 1.6}, and returns the exponent 1.6 as its output value. Here is the formal definition.
Definition: The natural logarithm is the function that takes any positive number x as input and returns the exponent to which the base e must be raised to obtain x. It is denoted ln(x). (e denotes the number 2.71828…) |
Note that to avoid confusion the natural logarithm function is denoted ln(x) and the base 10 logarithm function is denoted log(x) .
Example 1: Evaluate ln ( e^{ 4.7} ).
The argument of the natural logarithm function is already expressed as e raised to an exponent, so the natural logarithm function simply returns the exponent.
ln ( e^{ 4.7} ) = 4.7
Example 2: Evaluate ln ( 5 ).
According to the graph 5 = e^{ 1.6}. The logarithm function returns the exponent 1.6.
ln ( 5 ) = 1.6
Example 3: Evaluate ln ( e ).
Express the argument as e raised to the exponent 1 and return the exponent.
ln ( e ) = ln ( e^{ 1} ) = 1
Example 4: Evaluate ln ( 1 ).
Express the argument as e raised to the exponent 0 and return the exponent.
ln ( 1 ) = ln ( e^{ 0} ) = 0
Graph: The red curve shown to the right is the graph of the natural logarithm function, y = ln (x). Notice that for any positive x it is single valued and for any negative x it is undefined. If you compare this graph to the graph of y = e^{ x} then you see that one can be gotten from the other by interchanging the x and y axes.
For comparison the blue curve shows the base 10 logarithm function, y = log (x). It has exactly the same shape but is only 43% as tall.
Domain and range: The domain of the natural logarithm function is all positive real numbers and the range is all real numbers.
The natural logarithm function can be extended to the complex numbers, in which case the domain is all complex numbers except zero. The natural logarithm of zero is always undefined.
Solving the Equation e^{ x} = c for x by using the natural logarithm function
Suppose that x is unknown but that e^{ x} equals a known value c. Then finding x requires solving this equation for x:
e^{ x} = cThe solution is
x = ln (c)because finding ln (c) means expressing c as e raised to an exponent, and then returning that exponent. But the original equation says that that exponent is x. Note that if c is negative then there is no real solution. However there is a complex solution. Furthermore if c = 0 then there is no solution at all.
ln (x) and e^{ x} are inverse functions
Consider the e^{ x} function which takes x and returns e^{ x}, like this: The natural logarithm function is defined to do exactly the opposite, namely: Therefore these are inverse functions.
Note the following:
- We saw above that the solution of
e^{ x} = y is
x = ln (y). We should look at these two equations
as expressing the same relationship between x and y but
from different points of view. The first equation is the relationship solved for
y and the second one is the relationship solved for x.
(An analogy is that the statement “Tom is Jane’s brother”
is equivalent to the statement that “Jane is Tom’s sister”.)
- In the previous bullet we saw that the two equations,
e^{ x} = y and
x = ln (y), said the same thing.
If we replace x in the first equation
by the x of the second equation we get this identity:
e^{ ln (y)} = y
and if we replace y in the second equation by the y of the first equation we get this identity:x = ln (e^{ x })
These identities are useful for showing how the natural logarithm and e^{ x} functions cancel each other. - If you compare the graph
of y = ln (x) to the
graph
of y = e^{ x}
then you see that one can be gotten from the other by interchanging
the x and y axes. This always happens with inverse functions.
How to use the natural logarithm function in the Algebra Coach
- Type ln(x) into the textbox, where x is the argument.
The argument must be enclosed in brackets.
- Set the relevant options:
- Set the exact / floating point option. In floating point mode the natural logarithm of any number is evaluated. In exact mode the natural logarithm of an integer is not evaluated because to do so would result in an approximate number.
- Set the e does / does not represent 2.718… option. (Set this to does represent so that for example ln(e^{ 3 }) simplifies to 3.)
- Turn on complex numbers if you want to be able to evaluate the
natural logarithm of a negative or complex number.
- Click the Simplify button.
Algorithm for the natural logarithm function
Click here to see the algorithm that computers use to evaluate the natural logarithm function.
The exponential function (with base e, the e^{ x} function)
Background: You might find it useful to read the previous section on the natural logarithm function before reading this section. There we saw that it is possible to use the number e (which is approximately 2.71828…) as a base and to raise it to any power, x, and produce any positive number y. We can write this relationship in equation form:y = e^{ x}Here is a graph of y = e^{ x} (the blue curve). For comparison we also show graphs of y = 2^{ x} and y = 4^{ x}. Because the number e is between 2 and 4 the curve y = e^{ x} lies between the curves y = 2^{ x} and y = 4^{ x}.
All three of these curves are called exponential functions because the independent variable x is in the exponent. All three have the property that the higher up the curve you go the steeper they get. However y = e^{ x} has the special property that at every point along the curve the slope precisely equals the height. This property concerning slopes makes it a very important function in calculus.
Note: In this section when we say ‘the exponential function’ we mean the one with base e.
Definition: We define a function called the exponential function (denoted exp) that takes an argument x and returns the value of e raised to the power x: |
You can think of exp(x) as just an alternative (functional) notation for the exponential notation e^{ x}. So, of course, the functional form exp(x) has all the properties that the exponential form e^{ x} has. The Algebra Coach has an option that allows you to use one form or the other. Here is a table comparing the “look” of the various properties in the two forms:
Functional notation Exponential notation info on this property info on this property info on this property info on this property info on this property info on this property info on this property info on this property
Graph: The blue curve is the graph of y = e^{ x} (i.e. of the exponential function). It has the property that its slope equals its height everywhere. The dotted red lines show the slope of the curve at various points along the curve. Notice that the slope is 5 when the height is 5, and so on.
If you compare this graph of the exponential function to the graph of the natural logarithm function then you see that one can be gotten from the other by interchanging the x and y axes.
Domain and range: The domain of the exponential function is all real numbers and the range is all positive real numbers.
The exponential function can be extended to the complex numbers, in which case the domain and the range is all complex numbers.
Solving the Equation ln (x) = c for c by using the exponential function
Suppose that x is unknown but that ln (x) equals a known value c. Then finding x requires solving this equation for x:
ln (x) = c.The solution is:
x = e^{ c}, or x = exp (c).This was explained in the previous section on the natural logarithm function.
ln (x) and e^{ x} are inverse functions
This was explained in the previous section on the natural logarithm function.
How to use the exponential function in the Algebra Coach
- Type exp(x) into the textbox, where x is the argument.
The argument must be enclosed in brackets.
- Set the relevant options:
- Set the exponential function option to this:
- Set the exact / floating point option. In floating point mode exp(r) for any number r is evaluated. In exact mode exp(i) for any integer i is not evaluated because to do so would result in an approximate number.
- Turn on complex numbers if you want to be able to evaluate
exp(c) for a complex number c.
- Click the Simplify button.
Algorithm for the e^{ x} or exponential function
Click here to see the algorithm that computers use to evaluate e^{ x} or exp (x).
The square root function
Background: The area of a square is found by squaring the length of its side. If y is the area of the square and x is the length of a side, then the following formula gives the areay = x^{ 2}.This formula is fine if we know the length x and want to find the area y, but what if we know the area y and want to find the length x? Well, first we give the unknown length a name. We call it the square root (literally the answer to the square problem) (root in mathematics means answer). Then we create a square root function to calculate it. The input to the function is the area of the square and the output is the length of a side.
Here is a formal definition of the square root function:
Definition: The square root function is defined to take any positive number y as input and return the positive number x which would have to be squared (i.e. multiplied by itself), to obtain y. The square root of y is usually denoted like this: The symbol √ is called the radical symbol and the quantity inside it is called the argument of the square root. Note that in the Algebra Coach the square root of y must be typed in like this: sqrt (y). Some books denote the square root of y like this: √(y). |
Example 1: Evaluate . The argument, 16, of the square root function is easily expressed as 4^{ 2}, so the square root function returns the number 4. Example 2: Evaluate . In this example the argument of the square root function is already expressed as some number squared, so the square root function simply returns that number. Example 3: Evaluate . In this case the argument is a number which we don't know how to express as the square of some other number. Therefore we use a calculator or the Algebra Coach to evaluate the square root.
Graph: Here is a graph of the square root function.
Domain and range: The domain of the square root function is all non-negative real numbers and the range is all non-negative real numbers.
The square root function can be extended to the complex numbers, in which case the domain is all complex numbers.
and x^{ 2} are inverse functions
Click here for a review of inverse functions. The square function is the inverse of the square root function. However the square root function is the inverse of the square function only if the domain of the square function is restricted to the positive numbers.
The square root function is a one-to-one function that takes a non-negative number as input and returns the square root of that number as output. For example the number 9 gets mapped into the number 3. The square function takes any number (positive or negative) as input and returns the square of that number as output. For example the number 3 gets mapped into the number 9. Because the square function gives back the original number, it is the inverse of the square root function.
However the square function is a many-to-one function. For example both 3 and −3 get mapped into the number 9. Therefore the square function has no inverse. If it did, that mapping would be one-to-many and would not satisfy the ‘one range value’ requirement for a mapping to be a function.
This means that is always true for any x, but unless x happens to be a positive number. For example but .
Solving the equation x^{ 2} = y for x by using the square root function
Suppose that x is unknown but that x^{ 2} equals a known value y. Then finding x requires solving the following equation for x.
x^{ 2} = yThere are two solutions. One solution is This is because means the number which when squared would produce y. But the original equation says that this number is x.
There is a second solution. Because a negative number squared is positive, another solution is the negative of the first solution. These two solutions are usually put together using the plus or minus symbol (±) and expressed like this: This is spoken as “x equals plus or minus the square root of y”. Click here to see an alternative solution of the equation x^{ 2} = y that uses factoring.
Example 1: The solutions of the equation x^{ 2} = 16 are x = ± 4
Example 2: The solutions of the equation x^{ 2} = 5.7^{ 2} are x = ± 5.7
Example 3: The solutions of the equation x^{ 2} = −16 are the imaginary numbers x = ± 4 i
Example 4: The solutions of the equation x^{ 2} = −(5.7^{ 2 }) are the imaginary numbers x = ± 5.7 i
How to use the square root function in the Algebra Coach
- Type sqrt(x) into the textbox, where x is the argument.
The argument must be enclosed in brackets.
- Set the relevant options:
- Set the 1/2 exponent option to this:
- Set the exact / floating point option. In floating point mode the square root of any number is evaluated. In exact mode the square root of an integer is not evaluated if it would result in an approximate number.
- Turn on complex numbers if you want to be able to evaluate the
square root of a complex number.
- Click the Simplify button.
Algorithm for the square root function
Click here to see the algorithm that computers use to evaluate the square root function.
The power function
Before reading this section you may want to review the sections on exponents, where we explained what exponential notation is, and what it means for an exponent to be a negative number, zero, a fraction, or any real number in general.Definition: The power function is defined as the function that takes any number x as input, raises x to some power p, and returns x^{ p} as output. |
In a way this function takes two numbers, x and p, as input, but we will consider the power p to be a parameter (that is, a number which is held constant during the course of a problem but which may vary from problem to problem). This diagram shows the input x, the parameter p, and the output x^{ p}. Let the output of the power function be called y, so that
y = x^{ p}.Then, for example, if p = 2 then the power function becomes the so-called quadratic function y = x^{ 2}, and if p = 4 then the power function becomes the so-called quartic function y = x^{ 4}. We now investigate the value of the power function for various values of x and p.
Graph: The 3-dimensional plot to the right shows the power function for part of its domain. The variable x (the base) is plotted from left to right and the parameter p (the power) is plotted from front to back. The output value of the power function, x^{ p}, is plotted in the vertical direction. Thus the height of the surface gives the value of the power function.
For example in the front corner x = 0.25 and p = −4 so the height of the surface there is 0.25^{ −4} = 256, and in the back corner x = 4 and p = 4 so the height of the surface there is also 4^{ 4} = 256.
Domain and range:
Look at the picture to the right. This is the view looking straight down on the 3-D plot that was shown above. (The dotted rectangle is the part of the domain shown in the 3-D plot.) Everything shown in gray is part of the domain of the power function. This includes both the gray area on the right and the horizontal gray stripes on the left. Notice the following regions:
- If the base x is positive then any value of the power p
is possible. Thus the entire gray rectangle is part of the domain.
For this region the corresponding range is all positive real numbers.
- If the base x is zero then a negative power p would
result in division by zero. Thus the red line is not part of the domain.
However if p is zero or positive then there is no such problem.
So this is part of the domain and the corresponding range is just zero.
- If the base x is negative then only certain powers p
are possible:
- The power p can be any integer because this just means that you are
multiplying x by itself p times. For example is p
= 2 then the result is the quadratic function and when p
= 4 the result is the quartic function.
- The power p can be any fraction m/n whose denominator
n is odd, because x^{ m/n} just
means that you are taking the m^{ th} power of the
n^{ th} root of x, and the odd root
of a negative number is another negative number. For example:
Thus the gray horizontal lines are part of the domain of the power function.
The corresponding range is all real numbers except zero.
- The power p cannot be a fraction whose denominator is even because this would mean that you are taking the square root of a negative number. For example: The power p also cannot be an irrational number. Thus the spaces between the gray horizontal lines are not part of the domain of the power function.
- The power p can be any integer because this just means that you are
multiplying x by itself p times. For example is p
= 2 then the result is the quadratic function and when p
= 4 the result is the quartic function.
We can summarize all of the above points by stating that if the power p is a positive integer or a positive fraction with an odd denominator then the domain is all x. If the power p is a negative integer or a negative fraction with an odd denominator then the domain is all x except x = 0. For all other values of the power p the domain is positive x only.
The power function can be extended to the complex numbers, in which case the domain is all complex numbers x for all complex powers p, with one exception: x cannot equal zero if the power p is imaginary or has a negative real part. Click here to see why there is this exception.
The power functions with powers p and 1/p are inverses
If a function is one-to-one then it has an inverse. The power function x^{ p} is one-to-one if we restrict x to be positive and if the power p is not zero. This picture shows that its inverse is the power function x^{ 1/p}: The final simplification results from the properties of exponents. The power function x^{ 1/p} is often called the p^{th} root function.
If x is also allowed to be negative then the power function is one-to-one for certain powers p (e.g. odd powers such as p = 3), but many-to-one for others (e.g. even powers such as p = 2) and in those cases it does not have an inverse. For example with the result that both The problem is the second case where −3 doesn't get mapped back into −3.
The situation is even more complicated if x is allowed to be a complex number. Then the power function with odd powers becomes many-to-one as well. For example DeMoire's theorem shows that there are 3 complex numbers which when cubed give −8. If we then take the cube root of −8 there are two possible outcomes:
The result of taking the cube root depends on what mode the Algebra Coach is in. In exact mode it returns the number −2. In floating-point mode the fraction 1/3 becomes the floating point number 0.333… and the algorithm for evaluating the power function over the complex numbers is used and it returns the number 1 + 1.732 i.
Solving the Equation x^{ p} = c for x by using the power function
Suppose that x is unknown but that x^{ p} equals a known value c. Then finding x requires solving this equation for x:
x^{ p} = cThere are many cases, depending on what the power p is, whether c is positive or negative, whether we are looking only for a positive solution for x or all real solutions or all complex solutions. By far the simplest case is if c is positive and if we are only looking for a positive, real solution for x. Then there is only one solution and that solution is:
x = c^{ 1/p}.This follows from the fact that the power functions with powers p and 1/p are inverses.
If we are looking for all real solutions then the only other possible solution is a negative solution and the easiest way to find it is to just try the negative of previous solution. If it satisfies the equation then it is also a solution; if it doesn't then it is not. For example the positive solution of the equation x^{ 4} = 16 is x = 2, and if we try −2 we find that it is also a solution. On the other hand, the positive solution of the equation x^{ 3} = 8 is also x = 2, but if we try −2 then we find that it is not.
If c is negative then temporarily drop its − sign and find the one positive solution for x. Then attach a − sign to this solution and see if it satisfies the original equation. If it does then you have found the only real solution. If it doesn't then there is no real solution.
In all other cases the solution x = c^{ 1/p} will be complex and may be one of many possible complex solutions. Click here to see an example.
How to use the power function in the Algebra Coach
- Type x ^ p into the textbox, where x is base and p
is the power. Put brackets around x or p if required.
- Set the relevant options:
- Set the exact / floating point option. In floating point mode any power is evaluated. In exact mode a power is not evaluated if it would result in an approximate number.
- Set the 1/2 exponent option to this:
- Set the negative exponent option as desired.
- Turn on complex numbers if the base or power is complex or
if you expect the value returned by the power function to be a
complex number.
- Click the Simplify button.
Algorithm for the power function
Click here to see the algorithm that computers use to evaluate the power function.