1. Warm-up 3.3
Let and perform the
indicated operation. 1. 2. 3. 4.

2. 1. 2. 3. 4.

3. Inverse Functions 3.3B Standard: MM2A5 abcd
Essential Question: How do I graph and analyze exponential functions and their inverses?

4. Vocabulary
Inverse relation – A relation that interchanges the input and output value of the original relation
Inverse functions – The original relation and its inverse relation whenever both relations are functions
nth root of a – b is an nth root of a if bn = a
Horizontal Line Test – The inverse of a function f is also a function if and only if no horizontal line intersects the graph of f more than once

5. Example 1:
Graph y = 2x – 4 using a table and a red pencil.
Graph y = ½x + 2 using a table and a blue pencil. x y x y -4 -4 -2 1 1 -2 2 2 2 3 3 2 4 4 4 4

6. 8 6 4 2 -2 -4 -6 -8

7. x y x y -4 -4 -2 1 1 -2 2 2 2 3 3 2 4 4 4 4
a. What do you notice about the two tables?
The input (x) and output (y) are interchanged

8. b. With your pencil, draw the line y = x. 8 6 4 2 -2 -4 -6 -8
y = x
b. With your pencil, draw the line y = x.

9. What is the relationship between the red and blue lines and the line y = x?
The line y = x is the line of reflection for the graphs of the red and blue lines.
We say that y = 2x – 4 and y = ½x + 2 are inverse functions.
Let f(x) = 2x – 4 and f-1(x) = ½x + 2 .

10. To verify that f(x) = 2x – 4 and
are inverse functions you must show that
f(f-1(x)) = f-1(f(x)) = x.
= f(½x + 2)
= 2 (½x + 2) – 4
= x + 4 – 4
= x
= f-1(2x – 4)
= ½(2x – 4) + 2
= x – 2 + 2
= x

11. (2). Using composition of functions, determine if
f(x) = 3x + 1 and g(x) = ⅓x – 1 are inverse
functions?
f(g(x)) = f(⅓x – 1)
= 3(⅓x – 1) + 1
= x – 3 + 1
= x – 2
NO!

12. Example 3:
Graph y = x2 for x  using a table and a red pencil.
Graph y = √x using a table and a blue pencil. x y x y 1 1 1 1 2 4 4 2 3 9 9 3 4 16 16 4

13. 8 6 4 2 -2 -4 -6 -8

14. a. What do you notice about the two tables?
The input (x) and output (y) are interchanged x y x y 1 1 1 1 2 4 4 2 3 9 9 3 4 16 16 4

15. b. With your pencil, draw the line y = x. 8 6 4 2 -2 -4 -6 -8
b. With your pencil, draw the line y = x.

16. What is the relationship between the red and blue graphs and the line y = x?
The line y = x is the line of reflection for the graphs of the red and blue graphs.
We say that y = x2 and y = √x are inverse functions.
Let f(x) = x2 and f-1(x) = √x .

17. To verify that f(x) = x2 and
are inverse functions you must show that
f(f-1(x)) = f-1(f(x)) = x. =
= x =
= x

18. To find the inverse of a function that is one-to
one, interchange x with y and y with x, then solve
for y.
Find the inverse: y = 3x + 5
Inverse: x = 3y + 5
x – 5 = 3y

19. Find the inverse: y = 2x2 – 1
Inverse: x = 2y2 – 1
x + 1 = 2y2

20. Note: The function y = x2 is not a one-to-one function.
If the input and output were interchanged, the
graph of the new relation would NOT be a
y = x x = y2
So, we must restrict the
domain of functions that are not one-to-one in order to create a function with an inverse!

21. Sketch the graph of the inverse of each function.
6). x y 6 3 2 x y -4 -3 2

22. Sketch the graph of the inverse of each function.
7). x y -2 3 2 x y 6 -5 2