1. Section 2.6 Inverse Functions

2. Definition: Inverse The inverse of an invertible function f is the function f (read “f inverse”) where the ordered pairs of f are obtained by interchanging the coordinates in each ordered pair of f. The “x” values become the “y” values and vice versa. Please note that the number -1 is not an exponent when in reference to a function.

3. Example: Finding an inverse function Let f = {(1, 3), (4,2), (5, 7)} Find = {(3, 1), (2, 4),( 7, 5)} Find (3) (3) = 1 Find ( f )(1) = 1

4. Finding an inverse function. In an inverse function, the “x” values are switched with the “y” values to determine the inverse. Example: Let f(x) = 2x. That may be written as y = 2x. Switching, we obtain x = 2y. Solving for y to obtain the inverse, we get, and we write

5. Check domain and range As a last step, we check that the domain of f is the range of f inverse, and the range of f is the domain of f inverse.

6. Finding an inverse function. Let Find First, re-write original equation as

7. Then interchange x and y

8. Solve for y

10. Check domain and range Note that the domain of f is all real numbers except 3, and the range of f inverse is all real numbers except 3. Observe that has no solution.

11. Domain and Range Note that the domain of is the range of f, and the domain of f is the range of The “x” and “y” values are switched.

12. Definition: One to One A function f(x): X → Y is one to one if only when A one to one function has no two ordered pairs with different first coordinates and the same second coordinate. Example: is not one to one, As x and –x give the same y value

13. Theorem A function is invertible if and only if it is a one to one function.

14. Determining if a function is one to one Horizontal Line Test: If each horizontal line crosses the graph of a function at no more than one point, then the function is one to one.

15. Fails horizontal line test

16. Limiting the domain results in a function which passes the horizontal line test

17. Theorem A function g is the inverse of f if and only if: 1. The domain of g is equal to the range of f 2. for any x in the domain of f.

18. Inverse functions and graphing The graph of an inverse function is the reflection of the original function with respect to the line y = x.

19. Y=x

20. Assignment Given a function, find its inverse. Graph both, show that the inverse is a reflection of the original function across the line x=y