Power functions Functions of the form y = k * (x r ) where k is a non zero number and r is a positive integer are called power functions. Power functions have two basic shapes which you will investigate in this activity.

Inverse functions For many relationships which we describe with functions we have a choice of which variable is independent and which is dependent. Finding the inverse simply amounts to interchanging the independent and dependent variable.

Inverse functions If we a have a point (x, y) on a function then the point (y, x) lies on the inverse. If we have a table of values, the inverse is easy to find. The pair of tables below show a function and its inverse

Inverse functions The function f(x) = * x gives kilometers s a function of miles. If we interchange x and f(x) we have x = 1.609* f(x) which we can solve for f(x) to get f(x) =0.6215*x. This is the same relationship but this time kilometers are represented by x and f(x) represents the corresponding miles.

Inverse functions Given the graph of a function its inverse is simply a reflection about the line y = x.

Inverse functions We can verify the functions are inverses by first applying one then the other. This is called the composition of the two functions. Suppose f(x) = 1.609x and g(x) = x then f(g(x)) = 1.609(g(x)) = 1.609( x) = x so f and g are inverses.

Inverse of the exponential The exponential function f(x) = b x is very important in modeling. Thus its inverse denoted g(x) = log b (x) (read log base b of x) is also very important.

Inverse of the exponential To find values of the logarithm you find values of its inverse the exponential. For example, to find y = log 10 (1000) we solve 10 y = 1000 which gives y = 3.