1. Unit 1 Day 8 Inverse Functions
F-BF.4: Find an inverse function and use this relationship to sole problems using tables, graphs, and equations.

2. Recall: Definition of a function
In a function, each _______ can have only one ________
In other words, each ____ can only have one ____
Decide whether or not the relations below represent functions:
input
output x y
YES or NO
YES or NO
YES or NO
YES or NO
YES or NO

3. Inverse Functions opposite “undo” Square root 𝑥 𝑥 𝑦 domain and range
Inverse functions are ________________ or functions that ________________each other
Think of the function 𝑓 𝑥 = 𝑥 2 . How do you “undo” squaring x? ______________________
𝑥 2 and __________ are inverse functions
In inverse functions, the __ and __ values are switched from the original function
When x and y values switch places can you think of something else we have been learning about that might also switch? _____________________________________
When the x and y values switch, this results in a reflection over ____________
The inverse of the function f is labeled _______________. We read this as “f inverse.”
Square root 𝑥 𝑥 𝑦
domain and range
𝑦=𝑥
𝑓 −1 (𝑥)

4. f −1 (x)
For each table create a table that represents the inverse. Label the inverse correctly using function notation −4 −3 −2 −7 4 5
Does 𝑓(𝑥) represent a function? ____________
Does 𝑓 −1 (𝑥) represent a function? __________
Find 𝑓 0 : ______________________________________
What is 𝑓 −1 −3 ? _____________________________
What is 𝑓 −1 4 ? _____________________________
yes
yes −7 −2 5

5. h −1 (x)
For each table create a table that represents the inverse. Label the inverse correctly using function notation 9 −3 1 −1 1 1
yes
Does ℎ(𝑥) represent a function? ____________
Does ℎ −1 (𝑥) represent a function? __________
Find ℎ 0 : ___________________________________
What is ℎ −1 9 ? _____________________________
What is ℎ −1 0 ? _____________________________ no −3

6. Find the y-value in 𝑓 −1 in the table, then look for x
In the examples above, you were asked to evaluate the inverse function for a given input. Is there a pattern that you could use to evaluate the inverse of a function without creating an inverse table? __________________________________________________________________________________
If the point (−5, 3) is a point on 𝑓(𝑥), what point would be on 𝑓 −1 𝑥 ? _____________
If the point (8, 1) is a point on 𝑔(𝑥), what point would be on 𝑔 −1 𝑥 ? _______________
Find the y-value in 𝑓 −1 in the table, then look for x
(3, −5)
(1, 8)
Use the table of 𝒇(𝒙) below to answer the following questions:
𝑓 −1 9 =__________
𝑓 −1 (−2)= ________ −2 5

7. The function f(x) is shown on the graph
The function f(x) is shown on the graph. Using the same approach, used with the tables, find the values requested:
f −1 7 = _______
f −7 = _______
f −1 0 = _______
f 3 =_______
f −1 −3 = _______
f −1 −5 = _______ 5 −5 3 −7

8. Function Function Function Function Function
Vertical Line Test
To determine whether or not a graph represents a function we use the ______________________
If any vertical line touches the graph more than once, the graph is __________________
If any possible vertical line touches the graph only once, the graph is ________________
Determine whether or not the graphs below represent functions:
Vertical line test
not a function
a function
Function Function Function Function Function
Not a Function Not a Function Not a Function Not a Function Not a Function

9. Inverse from Graphs
To determine whether or not a functions inverse will also be a function use the ___________________
The same rules apply for an inverse function with the horizontal line test that apply for a function and the vertical line test
If a function does not pass the horizontal line test, it will have an inverse on a _________________
All ______________ functions will have an inverse function with a restricted domain along the line of symmetry
𝐡𝐨𝐫𝐢𝐳𝐨𝐧𝐭𝐚𝐥 𝐥𝐢𝐧𝐞 𝐭𝐞𝐬𝐭
𝐫𝐞𝐬𝐭𝐫𝐢𝐜𝐭𝐞𝐝 𝐝𝐨𝐦𝐚𝐢𝐧
𝐪𝐮𝐚𝐝𝐫𝐚𝐭𝐢𝐜

10. For each graph below, determine whether or not the inverse would represent a function. If an inverse does not exist, use vertical lines to create a domain where an inverse would exist.
Function Function Function Function
Not a Function Not a Function Not a Function Not a Function

11. Inverse Functions 𝐥𝐢𝐧𝐞𝐚𝐫 𝐬𝐪𝐮𝐚𝐫𝐞 𝐫𝐨𝐨𝐭 𝐜𝐮𝐛𝐞 𝐫𝐨𝐨𝐭
The inverse of a linear function is always a ____________ function
The inverse of a quadratic function is always a ______________ function
The inverse of a cubic function is always a _____________ function
𝐥𝐢𝐧𝐞𝐚𝐫
𝐬𝐪𝐮𝐚𝐫𝐞 𝐫𝐨𝐨𝐭
𝐜𝐮𝐛𝐞 𝐫𝐨𝐨𝐭

12. Finding the inverse from an equation:
Change the ____ to a ____
Switch the ____ and _____
Solve for _____
Use the notation ______ to represent your inverse
𝑓(𝑥) 𝑦 𝑥 𝑦 𝑦
𝑓 −1 (𝑥)

13. (1) Find the inverse of 𝑓 𝑥 =4𝑥−7
𝑦=4𝑥−7 𝑥=4𝑦−7
𝑥+7= 4𝑦
𝑓 −1 𝑥 = 𝑥+7 4

14. (2)Find the inverse of 𝑓 𝑥 =− 1 4 𝑥+8
𝑦=− 1 4 𝑥+8 𝑥=− 1 4 𝑦+8
− −8
− −4( )
𝑥−8 =
− 1 4 𝑦
𝑓 −1 𝑥 =−4𝑥+32

15. (3)Find the inverse of 𝑓 𝑥 =8 𝑥 2 −5
𝑦=8 𝑥 2 −5 𝑥=8 𝑦 2 −5
𝑥+5=
8 𝑦 2
𝑥+5 8 = y 2
𝑓 −1 𝑥 =± 𝑥+5 8
,𝑥≥−5

16. (4)Find the inverse of 𝑓 𝑥 = 𝑥+1 5 , 𝑥≥−1
𝑦= 𝑥+1 5
𝑥 = 𝑦+1 5
( )
𝑦+1
5𝑥=
25 𝑥 2 =
𝑦+1
− −1
𝑓 −1 𝑥 =25 𝑥 2 −1

17. (5)Find the inverse of 𝑓 𝑥 = 𝑥 −4
𝑦= 𝑥 −4 𝑥= 𝑦 −4
𝑥+4 = 𝑦
𝑓 −1 𝑥 = 𝑥+4 2