1. 5.3 Inverse Functions (Part I)

2. Objectives Verify that one function is the inverse function of another function. Determine whether a function has an inverse function.

3. Inverse functions have the effect of undoing each other. Inverse Functions Domain of f is the range of f -1 and vice versa.

4. Definition of Inverse Function A function g is the inverse of f if f(g(x)) = x for each x in the domain of g, and g(f(x)) = x for each x in the domain of f.

5. Important Observations 1.If g is the inverse of f, then f is the inverse of g. 2.The domain of f -1 is equal to the range of f. The range of f -1 is equal to the domain of f. 3.A function need not have an inverse; but if it does, the inverse is unique.

6. Examples Show that f(x) and g(x) are inverse functions.

7. Examples Show the f(x) and g(x) are inverse functions.

8. Reflective Property of Inverse Functions The graph of f contains the point (a,b) iff the graph of f -1 contains the point (b,a). The graph of f -1 is a reflection of the graph of f across the line y=x.

9. Existence of an Inverse Function 1.A function has an inverse iff it is one- to-one. 2.If f is strictly monotonic on its entire domain, then it is one-to-one and therefore has an inverse. Passes the horizontal line test, or it’s either always increasing or always decreasing on its domain.

10. Does the function have an inverse? (Can graph and see if it passes the horizontal line test or take the derivative and see if it changes sign in the domain.) Since the derivative is always > 0, the function has an inverse.

11. Does the function have an inverse? (Can graph and see if it passes the horizontal line test or take the derivative and see if it changes sign in the domain.) Since the derivative can be + or -, the function does not have an inverse.

12. Finding the Inverse of a Function 1.Make sure the function has an inverse. (horizontal line test or check the derivative) 2.Solve for x. 3.Switch x’s and y’s. 4.Domain of f -1 is the range of f. 5.Check f(f -1 (x))=x and f -1 (f (x))=x Note: #2 and 3 can be switched.

13. Find the inverse of f(x). f f -1 Solve for x. Switch x and y. Use f -1 (x) notation. Check your answer!

14. Find the inverse of f(x). Derivative can be + or – over the domain (all reals). Therefore, f(x) does not have an inverse. If you restrict the domain to [-π/2,π/2], it does have an inverse.

15. Homework 5.3 (page 338) #1-9 odd, 10, 11,12, 13-41 odd, 47-51 odd, 59-67 odd