1. Grade 11 University: (MCR3U) Unit 2: Transformation of Functions Domain and Range of Functions
Mr. Choi
© 2018 E. Choi – MCR3U - All Rights Reserved

2. Domain and Range
A function is a set of ordered pairs in which, for every x, there is only one y. The vertical line test is used to test for functions.
The domain is the set of first elements in the ordered pairs of a relation.
The range is All possible values of y (dependent variables) within the domain; the set of second elements in a relation.
Domain and Range of Functions
© 2018 E. Choi – MCR3U - All Rights Reserved

3. Recall: Six Basic (Parent) functions
Quadratic function
Radical function
𝒇 𝒙 = 𝒙 𝟐 x y -2 4  -1  1  0 1 2  4
𝒇 𝒙 = 𝒙 x y 0  1  1 4  2 9  3
𝑥|𝑥∈𝑅
𝑥|𝑥≥0𝑥∈𝑅
𝑦|𝑦≥0,𝑦∈𝑅
𝑦|𝑦≥0,𝑦∈𝑅
Cubic function
Absolute function
𝒇 𝒙 = 𝒙 𝟑 x y -2
-8  -1
 -1  0 1  1 2  8
𝒇 𝒙 =|𝒙| x y -2 2  -1  1  0 1 2  2
𝑥|𝑥∈𝑅
𝑦|𝑦∈𝑅
𝑥|𝑥∈𝑅
𝑦|𝑦≥0,𝑦∈𝑅
Domain and Range of Functions
© 2018 E. Choi – MCR3U - All Rights Reserved

4. Reciprocal (Linear) function
Recall: Six Basic (Parent) functions (Continued)
Reciprocal (Linear) function
𝒇 𝒙 = 𝟏 𝒙 x y -1 1 VA
x = 0 HA
y = 0
𝑥| 𝑥≠0,𝑥∈𝑅
𝑦| 𝑦≠0,𝑦∈𝑅
Exponential function (Next Unit)
𝒇 𝒙 = 𝟐 𝒙 x y -1
1/2   1 1  2
𝑥|𝑥∈𝑅
𝑦|𝑦>0,𝑦∈𝑅
Domain and Range of Functions
© 2018 E. Choi – MCR3U - All Rights Reserved

5. Example 1: Domain and Range of Linear Functions
Determine the domain and range for the following Linear Functions
D: 𝑥|𝑥∈𝑅
D: 𝑥|𝑥∈𝑅
R: 𝑦|𝑦∈𝑅
R: 𝑦|𝑦∈𝑅
D: 𝑥|𝑥∈𝑅
D: 𝑥|𝑥=3
R: 𝑦|𝑦=4
R: 𝑦|𝑦∈𝑅
Domain and Range of Functions
© 2018 E. Choi – MCR3U - All Rights Reserved

6. Quadratic Functions 𝒇 𝒙 = 𝒙 𝟐x y -2 4  -1  1  0 1 2  4
D: 𝑥|𝑥∈𝑅
R: 𝑦|𝑦≥0,𝑦∈𝑅
Domain and Range of Functions
© 2018 E. Choi – MCR3U - All Rights Reserved

7. Example 2: Domain and Range of Quadratic Functions
Determine the domain and range for the following Quadratic Functions
Prepare a quick image for the function
 Completing the squares
Vertex: (3, 6)
a > 0
Vertex: (-2, -9)
a > 0
D: 𝑥|𝑥∈𝑅
R: {𝑦|𝑦≥6, 𝑦∈𝑅}
D: 𝑥|𝑥∈𝑅
R: {𝑦|𝑦≥−9, 𝑦∈𝑅}
Domain and Range of Functions
© 2018 E. Choi – MCR3U - All Rights Reserved

8. Example 2: Domain and Range of Quadratic Functions
Determine the domain and range for the following Quadratic Functions
Prepare a quick image for the function
Vertex: (0, 3)
a < 0
Vertex: (0, 9)
D: 𝑥|𝑥∈𝑅
a < 0
R: {𝑦|𝑦≤3, 𝑦∈𝑅}
D: 𝑥|𝑥∈𝑅
R: {𝑦|𝑦≤9, 𝑦∈𝑅}
Domain and Range of Functions
© 2018 E. Choi – MCR3U - All Rights Reserved

9. Cubic Functions 𝒇 𝒙 = 𝒙 𝟑 D: 𝑥|𝑥∈𝑅 R: 𝑦|𝑦∈𝑅 𝒇 𝒙 = 𝒙 𝟑x y -2
-8  -1
 -1  0 1  1 2  8
D: 𝑥|𝑥∈𝑅
R: 𝑦|𝑦∈𝑅
Domain and Range of Functions
© 2018 E. Choi – MCR3U - All Rights Reserved

10. Example 3: Domain and Range of Cubic Functions
Determine the domain and range for the following Cubic Functions
Prepare a quick image for the function
Center: (2, 1)
Center: (-1, -1)
a > 0
a < 0
D: 𝑥|𝑥∈𝑅
D: 𝑥|𝑥∈𝑅
R: {𝑦|𝑦∈𝑅}
R: {𝑦|𝑦∈𝑅}
Domain and Range of Functions
© 2018 E. Choi – MCR3U - All Rights Reserved

11. Example 3: Domain and Range of Cubic Functions
Determine the domain and range for the following Cubic Functions
Prepare a quick image for the function
Center: (0, 2)
a < 0
Center: (4, 0)
a > 0
D: 𝑥|𝑥∈𝑅
R: {𝑦|𝑦∈𝑅}
D: 𝑥|𝑥∈𝑅
R: {𝑦|𝑦∈𝑅}
Domain and Range of Functions
© 2018 E. Choi – MCR3U - All Rights Reserved

12. Radical (Linear) Functions 𝒇 𝒙 = 𝒙
𝒇 𝒙 = 𝒙 x y 0  1  1 4  2 9  3
D: 𝑥|𝑥≥0𝑥∈𝑅
R: 𝑦|𝑦≥0,𝑦∈𝑅
Domain and Range of Functions
© 2018 E. Choi – MCR3U - All Rights Reserved

13. Example 4: Domain and Range of Radical (Linear) Functions
Determine the domain and range for the following Radical Linear Functions
Prepare a quick image for the function
Starting point: (1, 2)
Starting point: (-3, -2)
a > 0
k > 0
a < 0
k > 0
D: 𝑥|𝑥≥1,𝑥∈𝑅
D: 𝑥|𝑥≥−3,𝑥∈𝑅
R: {𝑦|𝑦≥2,𝑦∈𝑅}
R: {𝑦|𝑦≤−2,𝑦∈𝑅}
Domain and Range of Functions
© 2018 E. Choi – MCR3U - All Rights Reserved

14. Example 4: Domain and Range of Radical (Linear) Functions
Determine the domain and range for the following Radical Linear Functions
Prepare a quick image for the function
Starting point: (3, 1)
Starting point: (5, -3)
a > 0
k > 0
a < 0
k < 0
D: 𝑥|𝑥≥3,𝑥∈𝑅
D: 𝑥|𝑥≤5,𝑥∈𝑅
R: {𝑦|𝑦≥1,𝑦∈𝑅}
R: {𝑦|𝑦≤−3,𝑦∈𝑅}
Domain and Range of Functions
© 2018 E. Choi – MCR3U - All Rights Reserved

15. Absolute (Linear) Functions 𝒇 𝒙 =|𝒙|x y -2 2  -1  1  0 1 2  2
D: 𝑥|𝑥∈𝑅
R: 𝑦|𝑦≥0,𝑦∈𝑅
Domain and Range of Functions
© 2018 E. Choi – MCR3U - All Rights Reserved

16. Example 5: Domain and Range of Absolute Linear Functions
Determine the domain and range for the following Absolute Linear Functions
Prepare a quick image for the function
Vertex: (2, 4)
a > 0
Vertex: (-1, -3)
Vertex: (0, 3)
a < 0
a > 0
D: 𝑥|𝑥∈𝑅
D: 𝑥|𝑥∈𝑅
R: {𝑦|𝑦≥4, 𝑦∈𝑅}
D: 𝑥|𝑥∈𝑅
R: {𝑦|𝑦≤−3, 𝑦∈𝑅}
R: {𝑦|𝑦≥3, 𝑦∈𝑅}
Domain and Range of Functions
© 2018 E. Choi – MCR3U - All Rights Reserved

17. Reciprocal (Linear) Functions 𝒇 𝒙 = 𝟏 𝒙x y -1 1 VA
x = 0 HA
y = 0
𝑥| 𝑥≠0,𝑥∈𝑅
𝑦| 𝑦≠0,𝑦∈𝑅
Domain and Range of Functions
© 2018 E. Choi – MCR3U - All Rights Reserved

18. Example 6: Domain and Range of Reciprocal Linear Functions
Determine the domain and range for the following Reciprocal Linear Functions. Prepare a quick image for the function
Centre: (-1, -3)
Centre: (3, 2)
a > 0
a < 0
Centre: (0.5, 1)
a > 0
D: 𝑥|𝑥≠−1,𝑥∈𝑅
D: 𝑥|𝑥≠3,𝑥∈𝑅
R: {𝑦|𝑦≠−3, 𝑦∈𝑅}
R: {𝑦|𝑦≠2, 𝑦∈𝑅}
D: 𝑥|𝑥≠0.5,𝑥∈𝑅
R: {𝑦|𝑦≠1, 𝑦∈𝑅}
Domain and Range of Functions
© 2018 E. Choi – MCR3U - All Rights Reserved

19. Homework
Work sheet: Domain and Range of Functions Text: Check the website for updates
Domain and Range of Functions
© 2018 E. Choi – MCR3U - All Rights Reserved

20. End of lesson Domain and Range of Functions
© 2018 E. Choi – MCR3U - All Rights Reserved