1. Inverse Functions horizontal line test
Recall: To determine if a functions inverse is also a function use the ________________________
For each graph below, determine whether or not the inverse would represent a function
horizontal line test
YES or NO YES or NO YES or NO YES or NO

2. Inverse Functions Every restrict the domain quadratic, absolute value
_______ function has an inverse, but not every inverse will represent a function
For the inverse of a function to be a function, it may be necessary to ___________________
Think about the parent functions we have worked with this semester. Which function(s) would never pass the horizontal line test?
Both quadratic and absolute value functions will have inverse functions that exist on a restricted domain. To identify the domain, identify the _______ of the equation.
restrict the domain
quadratic, absolute value
vertex

3. Vertex at _______________ Inverse function will exist on the domain:
The function below does not pass the horizontal line test. Identify the vertex and use this point to restrict the domain so that the inverse will represent a function.
Vertex at _______________
Inverse function will exist on the domain:
__________ or __________
(−3,−4)
𝑥≤−3
𝑥≥−3

4. Vertex at _______________ Inverse function will exist on the domain:
The function below does not pass the horizontal line test. Identify the vertex and use this point to restrict the domain so that the inverse will represent a function.
Vertex at _______________
Inverse function will exist on the domain:
__________ or __________
(3,−2)
𝑥≤3
𝑥≥3

5. Vertex at _______________ Inverse function will exist on the domain:
The function below does not pass the horizontal line test. Identify the vertex and use this point to restrict the domain so that the inverse will represent a function.
Vertex at _______________
Inverse function will exist on the domain:
__________ or __________
(1,−3)
𝑥≤1
𝑥≥1

6. Vertex at _______________ Inverse function will exist on the domain:
The function below does not pass the horizontal line test. Identify the vertex and use this point to restrict the domain so that the inverse will represent a function.
Vertex at _______________
Inverse function will exist on the domain:
__________ or __________
(−1,4)
𝑥≤−1
𝑥≥−1

7. Restricted Domain for Inverse of Quadratics
As we just saw, when finding the restricted domain for the inverse of a quadratic function it helps to find the ___________
Recall vertex form: ______________________ where the vertex is the point ________
An inverse function would exist for the domain ____________
vertex
f x = x−h 2 +k
(h,k)
x≥h

8. Restricted Domain for Inverse of Quadratics
Given the functions below, identify the vertex and state the domain of the inverse:
𝑦= 𝑥−
Vertex at __________ Domain of inverse: _________
𝑦= 𝑥+4 2 −9
Vertex at __________ Domain of inverse: ___________
𝑦= 𝑥−
(5, 7)
𝑥≥5
𝑥≥−4
(−4, −9)
(3, 11)
𝑥≥3

9. Rearranging an equation into vertex form
To identify the vertex of a quadratic equation not in vertex form, ___________________ by forming a perfect square trinomial
Many quadratic equations need to be in _______________ to find the inverse
complete the square
vertex form

10. Rearrange the Quadratic Equation Below so that it is in vertex form
Rearrange the Quadratic Equation Below so that it is in vertex form. Then, identify a domain on which an inverse function would exist:
Find the number to create a perfect square trinomial.
− =
𝑥 2 −6𝑥+21=0 9
Add and subtract this number.
(𝑥 2 −6𝑥+9)−9+21=0
Re-write as perfect square and simplify.
𝑥−3 2
+12=0
Vertex at________
(3,12)
Inverse for ________
x≥3

11. Rearrange the Quadratic Equation Below so that it is in vertex form
Rearrange the Quadratic Equation Below so that it is in vertex form. Then, identify a domain on which an inverse function would exist:
Find the number to create a perfect square trinomial. =
𝑥 2 +10𝑥−11=0 25
Add and subtract this number.
(𝑥 2 +10𝑥+25)−25−11=0
Re-write as perfect square and simplify.
𝑥+5 2
−36=0
Vertex at________
(−5,−36)
Inverse for ________
x≥−5

12. Rearrange the Quadratic Equation Below so that it is in vertex form
Rearrange the Quadratic Equation Below so that it is in vertex form. Then, identify a domain on which an inverse function would exist:
Find the number to create a perfect square trinomial.
𝑥 2 +4𝑥+6=0 = 4
Add and subtract this number.
(𝑥 2 +4𝑥+4)−4+6=0
Re-write as perfect square and simplify.
𝑥+2 2
+2=0
Vertex at________
(−2,2)
Inverse for ________
x≥−2

13. Rearrange the Quadratic Equation Below so that it is in vertex form
Rearrange the Quadratic Equation Below so that it is in vertex form. Then, identify a domain on which an inverse function would exist:
Find the number to create a perfect square trinomial.
𝑥 2 −12𝑥−9=0
− = 36
Add and subtract this number.
(𝑥 2 −12𝑥+36)−36−9=0
Re-write as perfect square and simplify.
𝑥−6 2
−45=0
Vertex at________
(6,−45)
Inverse for ________
x≥6

14. Find the inverse of 𝑓 𝑥 = 𝑥 2 +4𝑥−9
𝑥 2 +4𝑥−9=0
𝑥 2 +4𝑥+4−4−9=0
𝑦+2 2 −13=𝑥
𝑥+2 2
−13=0
10000
10000
𝑥+2 2 =𝑥+13
𝑥+2=± 𝑥+13
−2 −2
𝑓 −1 𝑥 =−2± 𝑥+13

15. Find the inverse of 𝑦= 𝑥+3 2 −5
𝑥= 𝑦+3 2 −5
10000
10000
𝑥+5= 𝑦+3 2
± 𝑥+5 =𝑦+3
− −3
𝑓 −1 𝑥 =−3± 𝑥+5