Exercises: (+) Compose Functions

Work through each section in order. For composition, always identify the inner function first, then apply the outer function to the result.

Grade 9·21 problems·~28 min·Common Core Math - HS Functions·standard·hsf-bf-a-1c

Work through problems with immediate feedback

Warm-Up: Review What You Know

In the composition $f(g(x))$ , which function is applied first?

$f$ — it appears first reading left to right

$g$ — the inner function is evaluated first

Both are applied simultaneously

$f$ — it is on the outside, so it takes priority

Which of the following best describes what $f(g(x))$ means?

Multiply $f(x)$ and $g(x)$

Add $f(x)$ and $g(x)$

Use $g(x)$ as the input to $f$

Use $f(x)$ as the input to $g$

Using the values $f(3) = 7$ and $g(2) = 3$ , what is $f(g(2))$ ?

Fluency Practice

Let $f(x) = 2x + 1$ and $g(x) = x^2$ . Evaluate $f(g(3))$ .

Let $f(x) = 2x + 1$ and $g(x) = x^2$ . Evaluate $g(f(3))$ .

Let $f(x) = \sqrt{x}$ and $g(x) = x + 9$ . Find $f(g(x))$ and evaluate at $x = 7$ .

Let $f(x) = x^2$ and $g(x) = 3x - 1$ . Simplify $f(g(x))$ . What is the coefficient of $x^2$ in the simplified expression?

Let $f(x) = x + 5$ and $g(x) = 2x$ . Simplify $g(f(x))$ . What is the coefficient of $x$ in the simplified expression?

Word Problems

A weather balloon rises at a rate of $h(t) = 500t$ feet per hour (where $t$ is hours since launch). The temperature at height $y$ feet is $T(y) = 72 - 0.003y$ degrees Fahrenheit.

Write the composite function $T(h(t))$ that gives temperature as a function of time. Then interpret the meaning of the composition in one sentence.

A spherical balloon is being inflated. Its radius grows as $r(t) = 3t$ cm (where $t$ is seconds). The volume of a sphere is $V(r) = \frac{4}{3}\pi r^3$ .

Write the composition $V(r(t))$ as a function of time $t$ only. Simplify your answer.

Using your composition from part (a), find the volume (in cubic cm) after 2 seconds. Give your answer in terms of $\pi$ — enter the coefficient of $\pi$ .

The cost to produce $q$ items is $C(q) = 2q + 10$ dollars. The number of items produced in $t$ hours is $q(t) = 5t$ .

Find $C(q(t))$ — the cost as a function of time — and evaluate it at $t = 4$ hours.

Error Analysis

A student is given $f(x) = x + 1$ and $g(x) = x^2$ . The student writes: " $f(g(x)) = f(x) \cdot g(x) = (x+1)(x^2) = x^3 + x^2$ ."

What error did the student make?

The student multiplied $f(x) \cdot g(x)$ instead of substituting $g(x)$ into $f$

The student applied the functions in the wrong order

The algebra is correct

The student should have added $f(x) + g(x)$ instead

A student computes $f(g(x))$ and $g(f(x))$ for $f(x) = 2x$ and $g(x) = x + 3$ :

" $f(g(x)) = 2x + 3$ "
" $g(f(x)) = 2x + 3$ "
The student concludes: "Composition is commutative — the order doesn't matter."

Is the student's conclusion correct? Identify the error.

The student computed $f(g(x))$ incorrectly — it should be $2x + 6$ , not $2x + 3$

The student computed $g(f(x))$ incorrectly — it should be $2x + 6$ , not $2x + 3$

The student is correct — composition is always commutative

Both compositions are wrong — neither equals $2x + 3$

Challenge / Extension

Let $f(x) = \sqrt{x - 2}$ and $g(x) = x^2 + 2$ . What is the domain of $f(g(x))$ ?

All real numbers

$x \geq 2$

$x \geq 0$

$x \geq \sqrt{2}$

The function $h(x) = (2x - 3)^4$ can be decomposed in more than one way. Give two different decompositions as $h(x) = f(g(x))$ , using different inner and outer functions each time. Verify at least one decomposition by computing $f(g(x))$ .