Inverse Functions: Given a function, find its inverse.

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

2. 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.

3. 6 This means has domain [ 0,  ) and range [ - 1.5,  ). 5 Note has domain [ - 1.5,  ) and range [ 0,  ). This may be more apparent from its graph. 4 3 2 1 0 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 Inverse Functions: Given a function, find its inverse.

4. Inverse Functions: Given a function, find its inverse. Therefore, the domain restriction, [ 0,  ) must be included in the final answer:

5. 6 5 4 3 2 1 0 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 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. Note the graphs of f (x) and f -1(x) are symmetrical across the line, y = x. This is true of any function and its inverse.

6. The inverse function is: Inverse Functions: Given a function, find its inverse. Try: Given the one-to-one function, f (x) = x4 – 6, x 0, find f - 1(x).

7. Inverse Functions: Given a function, find its inverse. END OF PRESENTATION