Composition of Functions Practice

Calculate the inner output on a separate line, then feed it into the outer rule.

First \(g(5)=-1\). Use that complete output as the input of \(f\): \(f(-1)=2(-1)+3=1\).

Calculate the inner output on a separate line, then feed it into the outer rule.

First \(g(1)=-3\). Use that complete output as the input of \(f\): \(f(-3)=-3(-3)+2=11\).

Calculate the inner output on a separate line, then feed it into the outer rule.

First \(g(-2)=-4\). Use that complete output as the input of \(f\): \(f(-4)=4(-4)-1=-17\).

Calculate the inner output on a separate line, then feed it into the outer rule.

First \(g(3)=-5\). Use that complete output as the input of \(f\): \(f(-5)=5(-5)+2=-23\).

Calculate the inner output on a separate line, then feed it into the outer rule.

First \(g(-1)=-1\). Use that complete output as the input of \(f\): \(f(-1)=2(-1)-4=-6\).

Use parentheses around the full inner result before applying a power.

The inner value is \(g(4)=7\). Substituting the entire result into the square gives \(f(7)=(7)^2+3=52\).

Use parentheses around the full inner result before applying a power.

The inner value is \(g(-2)=5\). Substituting the entire result into the square gives \(f(5)=(5)^2-5=20\).

Use parentheses around the full inner result before applying a power.

The inner value is \(g(1)=5\). Substituting the entire result into the square gives \(f(5)=(5)^2+7=32\).

Use parentheses around the full inner result before applying a power.

The inner value is \(g(3)=-10\). Substituting the entire result into the square gives \(f(-10)=(-10)^2+1=101\).

Use parentheses around the full inner result before applying a power.

The inner value is \(g(-1)=-3\). Substituting the entire result into the square gives \(f(-3)=(-3)^2-2=7\).

Write f with an empty input slot, insert g(x) in parentheses, and simplify only afterward.

Replace every input in \(f\) with the complete expression \(g(x)\), then simplify. The resulting expression is \(3x+13\).

Write f with an empty input slot, insert g(x) in parentheses, and simplify only afterward.

Replace every input in \(f\) with the complete expression \(g(x)\), then simplify. The resulting expression is \(4x^2-4x+5\).

Write f with an empty input slot, insert g(x) in parentheses, and simplify only afterward.

Replace every input in \(f\) with the complete expression \(g(x)\), then simplify. The resulting expression is \(3-x^2\).

Write f with an empty input slot, insert g(x) in parentheses, and simplify only afterward.

Replace every input in \(f\) with the complete expression \(g(x)\), then simplify. The resulting expression is \(6x^2+7\).

Write f with an empty input slot, insert g(x) in parentheses, and simplify only afterward.

Replace every input in \(f\) with the complete expression \(g(x)\), then simplify. The resulting expression is \(x^2-8x+10\).

Read from the innermost parentheses outward; the rightmost named function acts first.

Because \(g\) is outside, substitute the entire formula \(2x+1\) for every input of \(g\). Correct expansion gives \(4x^2+4x-2\).

Read from the innermost parentheses outward; the rightmost named function acts first.

Because \(g\) is outside, substitute the entire formula \(x-4\) for every input of \(g\). Correct expansion gives \(3x-10\).

Read from the innermost parentheses outward; the rightmost named function acts first.

Because \(g\) is outside, substitute the entire formula \(x^2+1\) for every input of \(g\). Correct expansion gives \(3-x^2\).

Read from the innermost parentheses outward; the rightmost named function acts first.

Because \(g\) is outside, substitute the entire formula \(3x-5\) for every input of \(g\). Correct expansion gives \(18x^2-60x+50\).

Read from the innermost parentheses outward; the rightmost named function acts first.

Because \(g\) is outside, substitute the entire formula \(x+6\) for every input of \(g\). Correct expansion gives \(x^2+10x+24\).

Draw two short arrows—g then f, and f then g—before calculating.

Evaluating each pipeline independently gives \(f(g(4))=14\) and \(g(f(4))=18\). Their unequal values demonstrate that composition is generally order-dependent.

Draw two short arrows—g then f, and f then g—before calculating.

Evaluating each pipeline independently gives \(f(g(3))=4\) and \(g(f(3))=8\). Their unequal values demonstrate that composition is generally order-dependent.

Draw two short arrows—g then f, and f then g—before calculating.

Evaluating each pipeline independently gives \(f(g(-1))=-2\) and \(g(f(-1))=8\). Their unequal values demonstrate that composition is generally order-dependent.

Draw two short arrows—g then f, and f then g—before calculating.

Evaluating each pipeline independently gives \(f(g(2))=5\) and \(g(f(2))=15\). Their unequal values demonstrate that composition is generally order-dependent.

Draw two short arrows—g then f, and f then g—before calculating.

Evaluating each pipeline independently gives \(f(g(-2))=6\) and \(g(f(-2))=-6\). Their unequal values demonstrate that composition is generally order-dependent.

Circle the inner output and locate it in the input column before making the second lookup.

Read the inner function's row first, then use that output as the input row for the outer function. The two-step lookup gives \(f(g(-2))=-2\).

Circle the inner output and locate it in the input column before making the second lookup.

Read the inner function's row first, then use that output as the input row for the outer function. The two-step lookup gives \(g(f(0))=3\).

Circle the inner output and locate it in the input column before making the second lookup.

Read the inner function's row first, then use that output as the input row for the outer function. The two-step lookup gives \(f(g(4))=0\).

Circle the inner output and locate it in the input column before making the second lookup.

Read the inner function's row first, then use that output as the input row for the outer function. The two-step lookup gives \(g(f(-2))=0\).

Circle the inner output and locate it in the input column before making the second lookup.

Read the inner function's row first, then use that output as the input row for the outer function. The two-step lookup gives \(f(g(1))=1\).

Translate each ordered pair (input, output) into a lookup before composing.

Follow the ordered pair whose first coordinate matches the inner input, then use its second coordinate as the next input. This gives \(f(g(2))=5\).

Translate each ordered pair (input, output) into a lookup before composing.

Follow the ordered pair whose first coordinate matches the inner input, then use its second coordinate as the next input. This gives \(f(g(4))=6\).

Translate each ordered pair (input, output) into a lookup before composing.

Follow the ordered pair whose first coordinate matches the inner input, then use its second coordinate as the next input. This gives \(f(g(3))=1\).

Translate each ordered pair (input, output) into a lookup before composing.

Follow the ordered pair whose first coordinate matches the inner input, then use its second coordinate as the next input. This gives \(g(f(-4))=6\).

Translate each ordered pair (input, output) into a lookup before composing.

Follow the ordered pair whose first coordinate matches the inner input, then use its second coordinate as the next input. This gives \(g(f(9))=-1\).

Check the two domain gates in order: input into g, then g's output into f.

The composition is undefined because the inner output is not accepted by f. A composition requires the original input to work in \(g\) and the resulting output to work as an input of \(f\).

Check the two domain gates in order: input into g, then g's output into f.

The composition is undefined at the first stage. A composition requires the original input to work in \(g\) and the resulting output to work as an input of \(f\).

Check the two domain gates in order: input into g, then g's output into f.

The composition is defined and equals 9. A composition requires the original input to work in \(g\) and the resulting output to work as an input of \(f\).

Check the two domain gates in order: input into g, then g's output into f.

The composition is defined and equals -3. A composition requires the original input to work in \(g\) and the resulting output to work as an input of \(f\).

Check the two domain gates in order: input into g, then g's output into f.

The composition is undefined because the second lookup is missing. A composition requires the original input to work in \(g\) and the resulting output to work as an input of \(f\).

Write every intermediate output; do not multiply the number of applications by the first result.

Apply the same function exactly 3 times, carrying each output into the next application. The final value is \(11\).

Write every intermediate output; do not multiply the number of applications by the first result.

Apply the same function exactly 3 times, carrying each output into the next application. The final value is \(24\).

Write every intermediate output; do not multiply the number of applications by the first result.

Apply the same function exactly 2 times, carrying each output into the next application. The final value is \(8\).

Write every intermediate output; do not multiply the number of applications by the first result.

Apply the same function exactly 3 times, carrying each output into the next application. The final value is \(4\).

Write every intermediate output; do not multiply the number of applications by the first result.

Apply the same function exactly 4 times, carrying each output into the next application. The final value is \(15\).

Name the inner function and the intermediate output before judging a composition solution.

The function nearest x is g, so g must be evaluated first. This restores the inner-first pipeline and both required domain checks.

Name the inner function and the intermediate output before judging a composition solution.

The circle denotes composition, not multiplication. This restores the inner-first pipeline and both required domain checks.

Name the inner function and the intermediate output before judging a composition solution.

Every occurrence of u must be replaced by the entire expression g(x). This restores the inner-first pipeline and both required domain checks.

Name the inner function and the intermediate output before judging a composition solution.

Composition is generally order-dependent, so both pipelines must be calculated. This restores the inner-first pipeline and both required domain checks.

Name the inner function and the intermediate output before judging a composition solution.

The value 8 is only the inner output; without f(8), the composition is undefined. This restores the inner-first pipeline and both required domain checks.