PreCalculus 12

Chapter 4: Combining Functions Practice Test

Multiple Choice Identify the choice that best completes the statement or answers the question.

____ 1. Given , which pair of equations below are possible equations for and so that

?

A.

B.

C.

D.

____ 2. Given and , what is the domain of ?

A. C. B. D.

____ 3. Use the graphs of and . What are the domain and range of ?

0

y = g(x)y = f(x)

2 4 6–2–4–6 x

4

8

12

–4

–8

y

A. Domain: Range:

C. Domain: Range:

B. Domain: Range:

D. Domain: Range:

____ 4. Given , which pair of equations below are possible equations for and so that ?

A.

C.

B.

D.

PreCalculus 12

____ 5. Use these tables. What is the value of ?

x x

−3 16 −3 2 −2 9 −2 1 −1 4 −1 0 0 1 0 –1 1 0 1 –2 2 1 2 –3 3 4 3 –4

A. 1 C. –2 B. –1 D. 4

____ 6. The function is the composite of and . Which is an explicit equation for

, and what are the restrictions on x?

A.

or

C.

B.

D.

and

____ 7. For the functions and , which expression has the greatest value? A. B. C. D.

____ 8. Given the graphs of and , what is the value of ?

0

y = f(x)

y = g(x)

2 4 6 8–2–4–6–8 x

2

4

6

–2

–4

–6

–8

y

A. –5 B. –3.5 C. –2 D. –4.5

PreCalculus 12

____ 9. Given the functions and , what is the value of a for which ? A. 8 B. 4 C. –2 D. 0

____ 10. Given and , what is an explicit equation for ?

A. C. B. D.

____ 11. Given and , what is an explicit equation for ?

A. C. B. D.

PreCalculus 12

____ 12. Here are the graphs of and . Which graph below is the graph of ?

0

y = f(x)

y = g(x)4 8 12–4–8–12–16 x

4

8

12

16

–4

–8

–12

–16

y

A.

0 4 8 12–4–8–12–16 x

4

8

12

16

–4

–8

–12

–16

y C.

0 4 8 12–4–8–12–16 x

4

8

12

16

–4

–8

–12

–16

y

B.

0 4 8 12–4–8–12–16 x

4

8

12

16

–4

–8

–12

–16

y D.

0 4 8 12–4–8–12–16 x

4

8

12

16

–4

–8

–12

–16

y

PreCalculus 12

____ 13. Given and , what is an explicit equation for ? A. C. B. D.

____ 14. The function is the composite of and .

What is the domain of ? A. C. B. D. or

____ 15. Use these tables. What is the value of ?

x

−3 18 −2 11 −1 6 0 3 1 2 2 3 3 6

A. –2 C. 0 B. 2 D. 6

____ 16. Given and , what is an explicit equation for ? A. C. B. D.

____ 17. Given and , what is the domain and range of ?

A. Domain: Range:

C. Domain: Range:

B. Domain: Range:

D. Domain: Range:

PreCalculus 12

____ 18. Here are the graphs of and . Which graph below is the graph of ?

0

y = f(x)

y = g(x)

4 8 12–4–8–12 x

4

8

12

–4

–8

–12

y

A.

0 4 8 12–4–8–12 x

4

8

12

–4

–8

–12

y C.

0 4 8 12–4–8–12 x

4

8

12

–4

–8

–12

y

B.

0 4 8 12–4–8–12 x

4

8

12

–4

–8

–12

y D.

0 4 8 12–4–8–12 x

4

8

12

–4

–8

–12

y

PreCalculus 12

____ 19. Given and , what is an explicit equation for ?

A. C. B. D.

Short Answer

1. Given the functions and , determine an explicit equation for , then state

its domain. (2 marks)

2. Use the graphs of and .

a) State the domain and range of . b) State the domain and range of . c) Sketch the graph of .

(Make a table of values) d) What is the domain of ?

How is it related to the domain of and ?

Explain your work. (4 marks)

0

y = f(x)

y = g(x)

2 4 6–2–4–6–8–10 x

4

8

12

16

–4

–8

–12

–16

–20

y

3. Given the functions , , and , determine each value below. (1 mark each)

a)

b)

4. Given the functions , , and , determine an explicit equation for

, then state its domain. (2 marks)

PreCalculus 12

5. Consider the function and any function . Predict how the graph of each function below will be a transformation image of . (1 mark each) a) b) c)

d)

6. Given , , and , write an explicit equation for ,

then state its domain. (1 mark)

7. Given and , determine an explicit equation for each composite function, then state

its domain and range. Simplify if necessary. (1 mark each) a)

b) Show your work.

PreCalculus 12

Chapter 4: Combining Functions Practice Test Answer Section

MULTIPLE CHOICE

1. ANS: B PTS: 0 DIF: Easy

REF: 4.2 Combining Functions Algebraically LOC: 12.RF1 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge

2. ANS: D PTS: 0 DIF: Easy REF: 4.2 Combining Functions Algebraically LOC: 12.RF1 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge

3. ANS: D PTS: 0 DIF: Moderate REF: 4.1 Combining Functions Graphically LOC: 12.RF1 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge

4. ANS: A PTS: 0 DIF: Easy REF: 4.2 Combining Functions Algebraically LOC: 12.RF1 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge

5. ANS: D PTS: 0 DIF: Easy REF: 4.3 Introduction to Composite Functions LOC: 12.RF1 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge

6. ANS: D PTS: 0 DIF: Moderate REF: 4.4 Determining Restrictions on Composite Functions LOC: 12.RF1 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge

7. ANS: B PTS: 0 DIF: Moderate REF: 4.3 Introduction to Composite Functions LOC: 12.RF1 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge

8. ANS: C PTS: 0 DIF: Easy REF: 4.3 Introduction to Composite Functions LOC: 12.RF1 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge

9. ANS: C PTS: 0 DIF: Moderate REF: 4.3 Introduction to Composite Functions LOC: 12.RF1 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge

10. ANS: B PTS: 0 DIF: Easy REF: 4.2 Combining Functions Algebraically LOC: 12.RF1 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge

11. ANS: D PTS: 0 DIF: Easy REF: 4.2 Combining Functions Algebraically LOC: 12.RF1 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge

12. ANS: D PTS: 0 DIF: Easy REF: 4.1 Combining Functions Graphically LOC: 12.RF1 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge

13. ANS: A PTS: 0 DIF: Easy REF: 4.3 Introduction to Composite Functions LOC: 12.RF1 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge

14. ANS: A PTS: 0 DIF: Easy REF: 4.4 Determining Restrictions on Composite Functions LOC: 12.RF1 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge

PreCalculus 12

15. ANS: D PTS: 0 DIF: Easy REF: 4.3 Introduction to Composite Functions LOC: 12.RF1 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge

16. ANS: A PTS: 0 DIF: Easy REF: 4.3 Introduction to Composite Functions LOC: 12.RF1 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge

17. ANS: A PTS: 0 DIF: Moderate REF: 4.2 Combining Functions Algebraically LOC: 12.RF1 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge

18. ANS: A PTS: 0 DIF: Moderate REF: 4.1 Combining Functions Graphically LOC: 12.RF1 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge

19. ANS: B PTS: 0 DIF: Easy REF: 4.2 Combining Functions Algebraically LOC: 12.RF1 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge

SHORT ANSWER

1. ANS:

Domain: and

PTS: 0 DIF: Moderate REF: 4.4 Determining Restrictions on Composite Functions LOC: 12.RF1 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge

2. ANS: a) From the graph of : domain: ; range: b) From the graph of : domain: ; range: c) From the graphs, approximate values are:

x –6 6 0 0 –5 4 1.7 6.8 –4 2 2.4 4.8 –3 0 3.0 0.0 –2 –2 3.5 –7.0 –1 –4 3.9 –15.6

Plot the points at: (–6, 0), (–5, 6.8), (–4, 4.8), (–3, 0), (–2, –7), (–1, –15.6) Join the points with a smooth curve.

PreCalculus 12

0

y = f(x)

y = g(x)

y = f(x) g(x)•

2 4 6–2–4–6–8–10 x

4

8

12

16

–4

–8

–12

–16

y

d) The domain of is: The domain is the same as the domain of . It is not equal to the domain of , which is all

real numbers, because the graph of does not extend to the left of .

PTS: 0 DIF: Moderate REF: 4.1 Combining Functions Graphically LOC: 12.RF1 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge | Communication

3. ANS: a)

b)

PTS: 0 DIF: Moderate REF: 4.3 Introduction to Composite Functions LOC: 12.RF1 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge

4. ANS:

PTS: 0 DIF: Moderate REF: 4.4 Determining Restrictions on Composite Functions LOC: 12.RF1 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge

5. ANS: The function is a horizontal line with y-intercept . a) When is added to , the graph of will be translated 3 units down. b) When is subtracted from ,

PreCalculus 12

the graph of will be translated 3 units up. c) When is multiplied by , the graph of will be stretched vertically by a factor of 3 and reflected in the x-axis. d) When is divided by ,

the graph of will be compressed vertically by a factor of and

reflected in the x-axis.

PTS: 0 DIF: Moderate REF: 4.2 Combining Functions Algebraically LOC: 12.RF1 TOP: Relations and Functions KEY: Conceptual Understanding | Communication

6. ANS:

Domain:

PTS: 0 DIF: Moderate REF: 4.2 Combining Functions Algebraically LOC: 12.RF1 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge

7. ANS: a)

This is a cubic function; its domain is: ; its range is . b)

This is a cubic function; its domain is: ; its range is .

PTS: 0 DIF: Moderate REF: 4.3 Introduction to Composite Functions LOC: 12.RF1 TOP: Relations and Functions KEY: Conceptual Understanding | Procedural Knowledge | Communication