the square root function | the power function |
y = 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.
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). |
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.
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. |
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.
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.