practice problems: the composition of functions

Practice Problems: The Composition of FunctionsWork problems on your own first. Then check with answers in thefollowing slides. If the answers don’t help, complete solutions areavailable in the final slides.

1. Given 2)( xxf and xxg 3)( , find

a. ).5)(( gf b. )5)(( fg c. ))(( xgf d. ))(( xfg

2. Given 22)( xxxf and 43)( xxg , find

))((.

))((.

)2)((.

)2)((.

xfgd

xgfc

fgb

gfa

3. Given 24)( xxxf and 2)( xxg , find

a. )3)(( gf b. )3)(( fg c. ))(( xgf d. ))(( xfg

4. Given 6)( 2 xxf and xxg 5)( , find

))((.

))((.

)1)((.

)1)((.

xfgd

xgfc

fgb

gfa

5. Given xxf

2

2)( and

2)( xxg

))((.

))((.

)4)((.

)4)((.

xfgd

xgfc

fgb

gfa

6. Given 5)( 2xxf and 5)( xxg , find a. )6)(( gf b. )6)(( fg c. ))(( xgf d. ))(( xfg

7. Given 62)( xxf and xxg )( , find

))((.

))((.

))9((.

))9((.

xfgd

xgfc

fgb

gfa

8. Given xxf

2)( and xxg 4)(

))((.

))((.

))4((.

))4((.

xfgd

xgfc

fgb

gfa

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

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

Use the graphs of y = f(x) and y = g(x) to find each of the followingcompositions.

a. f(g(3))

b. g(f(3))

c. f(g(0))

d. g(f(0))

e. g(g(3))

Answers to Practice Problems for Composition of Functions(Complete solutions follow.)

1. a. 4 b. –22

2

2

3.

)3(.

xd

xc

463.

16269.

4.

0..2

2

2

xxd

xxc

b

a

2

2

11.

1.

10.

2..4

24.

224.

1.

3..3

xd

xc

b

a

xxd

xxc

b

a

Answers to Practice Problems for Composition of Functions(Complete solutions follow.)

xd

xc

b

a

xd

xc

b

a

.

.

6.

6..6

2

2.

2

2.

1.7

1..5

2

2

2

2

4.

4.

12.

0..8

26.

62.

6224.

0..7

xd

xc

b

a

xd

xc

b

a

9. a. –5 b. –1 c. 4 d. 2 e. 3

Complete solutions to Practice Problems for Composition ofFunctions.

463436

423)2())(()(.

16269162492

)43(2)43())(()(.

4)0())2(()2(.

044)2())2((2.

43)(,2)(.2

3)())(()(.

3)3())((.

22253)25())5((5.

2)2())5((5.

3)(,)(.1

22

22

22

2

2

22

2

2

2

xxxx

xxxxgxfgxfgd

xxxxx

xxxfxgfxgfc

gfgfgb

fgfgfa

xxgxxxf

xxgxfgxfgd

xxfxgfxgfc

gfgfgb

fgfgfa

xxgxxf

Complete solutions to Practice Problems for Composition ofFunctions.

222

2

22

1165)6())((.

165)5())((.

1055)5()1((1.

264)2())1((1.

5)(,6)(.4

244.

165)5(.

1123)3())3((3.

314)1())3((3..3

xxxgxfgxfgd

xxxfxgfxgfc

gfgfgb

fgfgfa

xxgxxf

xxxxgxfgd

xxxfxgfc

gfgfgb

fgfgfa

Complete solutions to Practice Problems for Composition ofFunctions.

xxxxgxfgxfgd

xxxfxgfxgfc

gfgfgb

fgfgfa

xxgxxf

xxxgxfgxfgd

xxfxgfxgfc

gfgfgb

fgfgfa

xxgx

xf

222

2

222

22

2

2

5)5()5())(())((.

5)5(5))(())((.

636)41())6(()6)((.

6)1())6(()6)((.

5)(,5)(.6

11656))(())((.

2

2)())(())((.

1)1()1())4(()4(.

7

1

162

2)16())4(()4)((.

)(,2

2)(.5

Complete solutions to Practice Problems for Composition ofFunctions.

22

2

2

4)())((.

4)4())((.

12164)16())4((.

0)0())4((.

4)(,)(.8

2662)62())((.

62)())((.

6264)24()24())9((.

0)3()9())9((.

5)(,5)(.7

xxgxfgd

xxfxgfc

gfgb

fgfa

xxgxxf

xxxgxfgd

xxfxgfc

gfgb

ffgfa

xxgxxf

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

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

Use the graphs of y = f(x) and y = g(x) to find each of the followingcompositions.

a. f(g(3))=

f(-1) = -5

b. g(f(3))=

g(3) = -1

c. f(g(0))=

f(2) = 4

d. g(f(0)) =

g(0) = 2

e. g(g(3) =

g(-1) = 3