1. Chapter 7 – Radical Equations and Inequalities 7.2 – Inverse Functions and Relations

2.  Today we will learn how to:  Find the inverse of a function or relation  Determine whether two functions or relations are inverses

3. 7.2 – Inverse Functions and Relations  Recall that a relation is a set of ordered pairs  Inverse Relation – set of ordered pairs obtained by reversing the coordinates of each ordered pair  The domain becomes the range  The range becomes the domain

4. 7.2 – Inverse Functions and Relations  Inverse Relation  Two relations are inverse relations if and only if whenever one relation contains the element (a, b), the other relation contains the element (b, a)  Q = {(1, 2), (3, 4), (5, 6)}  S = {(2, 1), (4, 3), (6, 5)}  Q and S are inverse relations

5. 7.2 – Inverse Functions and Relations  Example 1  The ordered pairs of the relation {(1, 3), (3, 6), (0, 6), (0, 1)} are the coordinates of the vertices of a rectangle. Find the inverse of the relation and determine whether the resulting ordered pairs are also the coordinates of the vertices of a rectangle.

6. 7.2 – Inverse Functions and Relations  The ordered pairs of inverse functions are also related  We write the inverse function of f(x) as f -1 (x)  Property of Inverse Functions  Suppose f and f -1 are inverse functions. Then, f(a) = b if and only if f -1 (b) = a

7. 7.2 – Inverse Functions and Relations  Let’s look at the inverse functions f(x) = x + 2 and f - 1 (x) = x – 2.  Evaluate f(5)  Since f(x) and f -1 (x) are inverses, f(5) = 7 and f -1 (7) = 5  The inverse function can be found by exchanging the domain and range of the function

8. 7.2 – Inverse Functions and Relations  Example 2  Find the inverse of f(x) = -1/2x + 1  Graph the function and its inverse

9. 7.2 – Inverse Functions and Relations  You can determine whether two functions are inverses by finding both of their compositions. If both equal the identity function f(x) = x, then the functions are inverse functions.  Two functions f and g are inverse functions is and only if both of their compositions are the identity function.  [ f ° g ]( x ) = x and [ g ° f ]( x ) = x

10. 7.2 – Inverse Functions and Relations  Example 3  Determine whether f(x) = 3/4x – 6 and g(x) = 4/3x + 8 are inverse functions.

11. 7.2 – Inverse Functions and Relations  You can also determine whether two functions are inverses by graphing.  Graphs of inverse functions are mirror images with respect to the graph of f(x) = x (identity function)

12. 7.2 – Inverse Functions and Relations  When the inverse of a function is a function, the original function is said to be one-to-one.  Just like the horizontal line test can determine if a graph is a function, the horizontal line graph is used to determine if the inverse of a function is a function itself.

13. 7.2 – Inverse Functions and Relations HOMEWORK Page 395 #11 – 35 odd, 36 – 38 all