Inverse Functions Given a function find its inverse

Inverse Functions: Given a function, find its inverse. Example: Given the one-to-one function, find f - 1(x). First rename f (x) as y. Second, interchange x and y. Third, solve for y. Table of Contents

Inverse Functions: Given a function, find its inverse. Fourth, rename y as f - 1(x). This is not the final result, however. Note something is wrong with the proposed inverse function found thus far, because it is not a one-to-one function. Recall the domain of a function is the range of its inverse. Also, the range of a function is the domain of its inverse. The last step is to determine if the inverse function found thus far needs to be altered to fit these facts. Table of Contents Slide 2

Inverse Functions: Given a function, find its inverse. Note has domain [ - 1. 5, ) and range [ 0, ). This may be more apparent from its graph. 6 5 4 3 2 1 This means has domain [ 0, ) and range [ - 1. 5, ). -6 -5 -4 -3 -2 -1 0 -1 -2 -3 -4 -5 -6 1 2 3 4 5 6 Therefore, the domain restriction, [ 0, ) must be included in the final answer: Table of Contents Slide 3

Inverse Functions: Given a function, find its inverse. Symmetry Between a Graph and Its Inverse The graphs of f (x), f -1(x) of the preceding example are shown along with the graph of y = x. 6 5 4 3 2 1 -6 -5 -4 Note the graphs of f (x) and f -1(x) are symmetrical across the line, y = x. -3 -2 -1 0 -1 -2 -3 -4 -5 -6 1 2 3 4 5 6 This is true of any function and its inverse. Table of Contents Slide 4

Inverse Functions: Given a function, find its inverse. Try: Given the one-to-one function, f (x) = x 4 – 6, x 0, find f - 1(x). The inverse function is: Table of Contents Slide 5

Inverse Functions: Given a function, find its inverse. Table of Contents