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: |
If this function is called f then f (10) = 3 and f (2) = 15.
domain range 10 3 2 15 5 46
h (x) = x^{ 2} − 2 xThis 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.
y = x^{ 2} − 2 x.In this form h (x) has been replaced by a new variable y so that there are now two variables, x and y. Variable y is the value of the range that corresponds to the value of variable x of the domain.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.
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.
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.
Algebra Coach Exercises |
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.
Algebra Coach Exercises |
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.
Algebra Coach Exercises |