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Precalculus : Composition of Functions

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Example Questions

Example Question #1 : Composition Of Functions

Suppose f(x) = 9 - x^3 and g(x) = ln (x+1)

What would g(f(x)) be?

Possible Answers:

9 - ln^2(x+1)

ln (9-x^3)

ln (10 - x^3)

10 - ln^3(x + 1)

ln(10 + x - x^3)

Correct answer:

ln (10 - x^3)

Explanation:

Substitute f(x) = 9 -x^3 into the function g(x) = ln (x+1) for .

Then it will become:

g(9 - x^3) = ln ((9-x^3) + 1) = ln (10 - x^3)

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Example Question #2 : Composition Of Functions

$f(x)=3-2x^2+\ln(x)$

g(x)=e^x

What is f(g(x)) ?

Possible Answers:

$3-2e^{2x} +x$

2x^2 +x

3-2e^x +x

3-2x^2+e^x

$3-2e^{2x} +\ln(2e^x)$

Correct answer:

$3-2e^{2x} +x$

Explanation:

f(g(x)) simply means: where ever you see an x in the equation f(x), replace it with g(x).

So, doing just that, we get

$3-2(e^x)^2+\ln(e^x)$ ,

which simplifies to

$3-2e^{2x}+x\ln(e)$ .

Since

$\ln(e)=1$ our simplified expression becomes,

$3-2e^{2x} +x$ .

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Example Question #2 : Composition Of Functions

f(x)=2-5x

$g(x)=\frac{3x^2}{x}-6$

What is g(f(x)) ?

Possible Answers:

-15x

225x^2 +6

12-15x

$\frac{75x}{4}$

-15x+32

Correct answer:

-15x

Explanation:

g(f(x)) simply means replacing every x in g(x) with f(x).

$\frac{3(2-5x)^2}{(2-5x)}-6$

After simplifying, it becomes

$\frac{3(2-5x)(2-5x)}{(2-5x)}-6$

=3(2-5x)-6

=6-15x-6=-15x

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Example Question #3 : Composition Of Functions

For the functions

f(x)=4x

and

g(x)=x^2 .

Evaluate the composite function

$f\circ g$ .

Possible Answers:

DNE

4x^2

8x^3

16x^2

Correct answer:

4x^2

Explanation:

The composite function means to plug in the function of g(x) into the function f(x) for every x value in the function.

Therefore the composition function becomes:

$f\circ g = f[g(x)] = 4(x^2) = 4x^2$ .

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Example Question #4 : Composition Of Functions

For the functions

f(x)=x+4

and

g(x)=x^2 .

Evaluate the composite function

$f\circ g$ .

Possible Answers:

x^2+4

DNE

x^2+x+4

x^2+8x+16

Correct answer:

x^2+4

Explanation:

The composite function means to plug in the function g(x) into f(x) for every x value.

Therefore the composite function becomes,

$f\circ g = f[g(x)] = (x^2)+4 = x^2+4$

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Example Question #4 : Composition Of Functions

If f(x)=x^2-3 , g(x)=6x+4 , and h(x)=x+8 , what is f(g(h(-7))) ?

Possible Answers:

288

Correct answer:

Explanation:

When doing a composition of functions such as this one, you must always remember to start with the innermost parentheses and work backward towards the outside.

So, to begin, we have

h(-7)=(-7)+8=1 .

Now we move outward, getting

g(h(-7))=g(1)=6(1)+4=10 .

Finally, we move outward one more time, getting

f(g(h(-7)))=f(g(1))=f(10)=(10)^2-3=97 .

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Example Question #3 : Composition Of Functions

Find g(h(i(2))) if g(x) = x , h(x) = 3 , and i(x)=x^4 .

Possible Answers:

3+x

48x

Correct answer:

Explanation:

Solve for the value of i(2) .

i(2)= 2^4=16

Solve for the value of h(i(2)) .

h(i(2))=h(16) = 3

Solve for the value g(h(i(2))) .

g(h(i(2)))=g(3) = 3

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Example Question #8 : Composition Of Functions

For the functions f(x)=x+7 and $g(x)=x^{3}$ , evaluate the composite function $f\circ g$

Possible Answers:

$(x+7)^{3}$

$x^{3}+7x^{2}+14x+21$

$x^{3}+21x^{2}$

$x^{3}+49x^{2}$

$x^{3}+7$

Correct answer:

$x^{3}+7$

Explanation:

The composite function notation $f\circ g$ means to swap the function g(x) into f(x) for every value of . Therefore:

$f\circ g$

=f[g(x)]

$=(x^{3})+7$

$=x^{3}+7$

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Example Question #5 : Composition Of Functions

For the functions f(x)=x-7 and g(x)=4x+6 , evaluate the composite function $g\circ f$ .

Possible Answers:

3x+13

3x-1

4x-22

4x-1

4x-13

Correct answer:

4x-22

Explanation:

The composite function notation $g\circ f$ means to swap the function f(x) into g(x) for every value of . Therefore:

$g\circ f$

=g[f(x)]

=4(x-7)+6

=4x-28+6

=4x-22

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Example Question #10 : Composition Of Functions

For the functions $f(x)=x^{2}-4$ and g(x)=2x+3 , evaluate the composite function $f\circ g$ .

Possible Answers:

$4x^{2}+12x+5$

$2x^{2}-4$

$4x^{2}+12x+9$

None of the answers listed

$2x^{2}-1$

Correct answer:

$4x^{2}+12x+5$

Explanation:

The composite function notation $f\circ g$ means to swap the function g(x) into f(x) for every value of . Therefore:

$f\circ g$

=f[g(x)]

$=(2x+3)^{2}-4$

$=4x^{2}+12x+9-4$

$=4x^{2}+12x+5$

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Precalculus : Composition of Functions

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Composition Of Functions

Example Question #2 : Composition Of Functions

Example Question #2 : Composition Of Functions

Example Question #3 : Composition Of Functions

Example Question #4 : Composition Of Functions

Example Question #4 : Composition Of Functions

Example Question #3 : Composition Of Functions

Example Question #8 : Composition Of Functions

Example Question #5 : Composition Of Functions

Example Question #10 : Composition Of Functions

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