Curriculum 2.0 Algebra 2: Unit 1-Topic 2, SLT 6
Name:
A Quadratic and Its Inverse Sample Answers
Date:
Period:
1. Graph f ( x)  x 2 on the coordinate grid.
2. On the same grid, graph the relation that represents the inverse of f.
3. How are the graphs related?
The graphs are inverses of one another. Every corresponding pair of coordinates follows the
relationship ( x, y )  ( y, x) .
4. Determine the domain and range of f.
Domain:  ,   ,    x   , or R
Range: [0,) or y  0
5. Determine the domain and range of the inverse of f.
Domain: [0,) or x  0
Range:  ,  ,    y   , or R
6. Is the inverse of f a function? Why or why not?
f ( x)  x 2 is a function, as for every input, there is exactly one output. Its inverse is not a function, as
there are multiple outputs for a given input (except 0).
Page 1 of 4
Curriculum 2.0 Algebra 2: Unit 1-Topic 2, SLT 6
Name:
A Quadratic and Its Inverse Sample Answers
Date:
Period:
7. If we restrict the domain of f ( x)  x 2 to x  0 , will its inverse be a function?
Yes. This will restrict the inverse function’s range to y-values greater than or equal to 0, so each input
will have exactly one output.
8. Graph f ( x)  x 2 , with restricted domain x  0 on the coordinate grid.
9. Write the equation for f 1 using what you know about inverse operations.
f 1 ( x )  x
10. On the same grid, graph the function that represents the inverse of f.
11. Determine the domain and range of f 1 .
Domain: [0,) or x  0
Range: [0,) or y  0
12. On which intervals is f 1 increasing? Decreasing?
Increasing: [0,) or x  0
Decreasing: Never
Page 2 of 4
Curriculum 2.0 Algebra 2: Unit 1-Topic 2, SLT 6
Name:
A Quadratic and Its Inverse Sample Answers
Date:
Period:
13. Graph f ( x)  x 2  3 on the coordinate grid.
14. On the same grid, graph the relation that represents the inverse of f.
15. What restrictions can be placed on the original function to ensure that its inverse is also a function?
If we restrict the domain of f ( x)  x 2  3 to x  0 , its inverse will be a function.
Note: There are a variety of ways in which students can restrict the domain of the original function to
ensure that the inverse is also a function. For example, students may choose to restrict the domain of
f ( x)  x 2  3 to x  0 . Consider discussing other ways in which the domain of the original function
could be restricted, and what implications the restriction would have on the inverse function and its key
characteristics. Make connections between the domain restriction and the function characteristics in
#16–18.
16. Determine the domain and range of f 1 .
Domain: [3,) or x  3
Range: [0,) or y  0
17. On which intervals is f 1 increasing? Decreasing?
Increasing: [3,) or x  3
Decreasing: Never
18. What is the left endpoint of the graph of f 1 ?
(3, 0)
Page 3 of 4
Curriculum 2.0 Algebra 2: Unit 1-Topic 2, SLT 6
Name:
A Quadratic and Its Inverse Sample Answers
Date:
Period:
19. Graph f ( x)  ( x  3)2 on the coordinate grid.
20. On the same grid, graph the relation that represents the inverse of f.
21. What restrictions can be placed on the original function to ensure that its inverse is also a function?
If we restrict the domain of f ( x)  ( x  3)2 to x  3 , its inverse will be a function.
Note: There are a variety of ways in which students can restrict the domain of the original function to
ensure that the inverse is also a function. For example, students may choose to restrict the domain of
f ( x)  ( x  3)2 to x  3 . Consider discussing other ways in which the domain of the original function
could be restricted, and what implications the restriction would have on the inverse function and its key
characteristics. Make connections between the domain restriction and the function characteristics in
#22–24.
22. Determine the domain and range of f 1 .
Domain: [0,) or x  0
Range: [  3,) or y  3
23. On which intervals is f 1 increasing? Decreasing?
Increasing: [0,) or x  0
Decreasing: Never
24. What is the left endpoint of f 1 ?
(0, 3)
Page 4 of 4