Power functions

•Functions of the form y = k * (xr) 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

Miles Kilometers1 1.6095 8.045

10 16.0915 24.135

Kilometers Miles1.609 18.045 516.09 10

24.135 15

Inverse functions

•The function f(x) = 1.609 * 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) =0.621504x then f(g(x)) = 1.609(g(x)) = 1.609(0.621504x) = x so f and g are inverses.

Inverse of the exponential

•The exponential function f(x) = bx is very important in modeling. •Thus its inverse denoted g(x) = logb(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 = log10 (1000) we solve 10y = 1000 which gives y = 3.