Precalculus : Composition of Functions

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #1 : Composition Of Functions

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

What would \(\displaystyle g(f(x))\) be?

Possible Answers:

\(\displaystyle ln(10 + x - x^3)\)

\(\displaystyle 10 - ln^3(x + 1)\)

\(\displaystyle ln (10 - x^3)\)

\(\displaystyle ln (9-x^3)\)

\(\displaystyle 9 - ln^2(x+1)\)

Correct answer:

\(\displaystyle ln (10 - x^3)\)

Explanation:

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

Then it will become:

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

Example Question #1 : Composition Of Functions

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

\(\displaystyle g(x)=e^x\)

What is \(\displaystyle f(g(x))\)?

Possible Answers:

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

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

\(\displaystyle 3-2e^x +x\)

\(\displaystyle 3-2x^2+e^x\)

\(\displaystyle 2x^2 +x\)

Correct answer:

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

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 

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

which simplifies to 

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

Since 

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

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

Example Question #2 : Composition Of Functions

\(\displaystyle f(x)=2-5x\)

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

What is \(\displaystyle g(f(x))\)?

Possible Answers:

\(\displaystyle 12-15x\)

\(\displaystyle \frac{75x}{4}\)

\(\displaystyle 225x^2 +6\)

\(\displaystyle -15x+32\)

\(\displaystyle -15x\)

Correct answer:

\(\displaystyle -15x\)

Explanation:

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

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

After simplifying, it becomes

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

\(\displaystyle =3(2-5x)-6\)

\(\displaystyle =6-15x-6=-15x\)

Example Question #1 : Composition Of Functions

For the functions

\(\displaystyle f(x)=4x\)

and

\(\displaystyle g(x)=x^2\).

Evaluate the composite function

\(\displaystyle f\circ g\).

Possible Answers:

DNE

\(\displaystyle 8x^3\)

\(\displaystyle 4x^2\)

\(\displaystyle 16x^2\)

Correct answer:

\(\displaystyle 4x^2\)

Explanation:

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

Therefore the composition function becomes:

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

Example Question #2 : Composition Of Functions

For the functions

\(\displaystyle f(x)=x+4\)

and

\(\displaystyle g(x)=x^2\).

Evaluate the composite function

\(\displaystyle f\circ g\).

Possible Answers:

\(\displaystyle x^2+x+4\)

DNE

\(\displaystyle x^2+8x+16\)

\(\displaystyle x^2+4\)

Correct answer:

\(\displaystyle x^2+4\)

Explanation:

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

Therefore the composite function becomes,

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

Example Question #2 : Composition Of Functions

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

Possible Answers:

\(\displaystyle 46\)

\(\displaystyle -38\)

\(\displaystyle 97\)

\(\displaystyle 288\)

\(\displaystyle 1\)

Correct answer:

\(\displaystyle 97\)

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

 \(\displaystyle h(-7)=(-7)+8=1\).

Now we move outward, getting 

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

Finally, we move outward one more time, getting 

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

Example Question #3 : Composition Of Functions

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

Possible Answers:

\(\displaystyle 3+x\)

\(\displaystyle 3\)

\(\displaystyle 16\)

\(\displaystyle 48\)

\(\displaystyle 48x\)

Correct answer:

\(\displaystyle 3\)

Explanation:

Solve for the value of \(\displaystyle i(2)\).

\(\displaystyle i(2)= 2^4=16\)

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

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

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

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

Example Question #1 : Composition Of Functions

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

Possible Answers:

\(\displaystyle x^{3}+7\)

\(\displaystyle x^{3}+21x^{2}\)

\(\displaystyle (x+7)^{3}\)

\(\displaystyle x^{3}+7x^{2}+14x+21\)

\(\displaystyle x^{3}+49x^{2}\)

Correct answer:

\(\displaystyle x^{3}+7\)

Explanation:

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

\(\displaystyle f\circ g\)

\(\displaystyle =f[g(x)]\)

\(\displaystyle =(x^{3})+7\)

\(\displaystyle =x^{3}+7\)

Example Question #2 : Composition Of Functions

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

Possible Answers:

\(\displaystyle 4x-13\)

\(\displaystyle 3x+13\)

\(\displaystyle 3x-1\)

\(\displaystyle 4x-22\)

\(\displaystyle 4x-1\)

Correct answer:

\(\displaystyle 4x-22\)

Explanation:

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

\(\displaystyle g\circ f\)

\(\displaystyle =g[f(x)]\)

\(\displaystyle =4(x-7)+6\)

\(\displaystyle =4x-28+6\)

\(\displaystyle =4x-22\)

Example Question #1 : Composition Of Functions

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

Possible Answers:

\(\displaystyle 2x^{2}-1\)

\(\displaystyle 4x^{2}+12x+9\)

\(\displaystyle 2x^{2}-4\)

\(\displaystyle 4x^{2}+12x+5\)

None of the answers listed

Correct answer:

\(\displaystyle 4x^{2}+12x+5\)

Explanation:

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

\(\displaystyle f\circ g\)

\(\displaystyle =f[g(x)]\)

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

\(\displaystyle =4x^{2}+12x+9-4\)

\(\displaystyle =4x^{2}+12x+5\)

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