Warm-up: Use the graph to evaluate each function. 1. 2. 3. 4. 5. 6. g(x) ● ● ● f(x) ● ● Answers: (1) -9 (2) 0 (3) -2 (4) -3 (5) undefined (6) 2

Warm-up KEY: Use the graph to evaluate each function. 1. 2. 3. 4. 5. 6. = -9 g(x) = 0 ● ● ● f(x) = -2 = -3 ● ● Answers: (1) -9 (2) 0 (3) -2 (4) -3 (5) undefined (6) 2 = 2 = 2/0 = und

HW Key:

Objectives & HW: Objectives: Homework: Students will be able to find, evaluate, and graph inverse functions. Inverses of Functions WKS

Inverse Relations… Example…Find the inverse of the relation. Two relations are inverse relations if and only if whenever one relation contains the element (a, b) and the other relation contains the element (b, a). Example…Find the inverse of the relation. State whether the relation is a function and whether its inverse is a function. no; x-values repeat Function(yes or no) inverse: yes; x-values don’t repeat Function(yes or no)

Inverse Functions… Example…Find the inverse of Suppose f and f-1 are inverse functions. Then f(a) = b if and only if f(b) = a. {Note: the -1 is not an exponent} Example…Find the inverse of Replace f(x) with y Interchange x and y Solve for y Replace y with f-1

Practice: Find the inverse of each function. 1. 2.

Definition… Example…Prove that g(x) and h(x) are not inverses. Words…Two functions f and g are inverse functions if and only if both of their compositions are the identity function. Symbols… Example…Prove that g(x) and h(x) are not inverses. Therefore, g and h are not inverses.

Inverse Functions… Definition…When the inverse of a function is a function, then the original function is said to be one-to- one. To determine if the inverse of a function is a function, you can use the horizontal line test. Example…Graph the function and determine if its inverse is a function. g-1(x) is/is not a function

Practice… 3. Graph and its inverse on the same plane. The function and its inverse will be reflections over the identity function y = x

Non-linear functions… Find the inverse of each function. 1. 2.

Non-linear functions… Find the inverse of each function. 3. 4. write on board about why you need + and – in the inverse…talk about x^2=16 and why the answer is both + and – 4 and also why the inverse of a parabola is not a function…if you just have sq.rt.x that is a function so you need both

Find the inverse of each function. 5. 6.

7. Prove that are not inverses. 8. Prove that are inverses. Inverse Proof…Use composition to determine if the pair of functions are inverses. 7. Prove that are not inverses. 8. Prove that are inverses. Therefore, f and g are not inverses. Therefore, h and k are inverses.

Practice… 9. Graph and its inverse on the same plane. The function and its inverse will be reflections over the identity function y = x

Try the last few problems… Think of them as extensions and challenge problems. They could be asked as bonus questions on another assignment.