Knowledge Notes | Carey AP Precalculus Practice Lab

Feed the inner output into the outer function, and check both domains.

What a strong response shows

The rightmost, inner function acts first.
Use its output as the outer function input.
Require the original input and intermediate output to be allowed.

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What You Need to Know

Key idea

Composition feeds the output of one function into another. The rightmost function acts first.

Fast decision rule

Start at the input, apply the inner function, then use that result as the outer input.

Worked steps

For a two-stage process, first compute the intermediate value, then apply the second rule to that value.

Common mistake

Reversing the order because the written expression looks like ordinary multiplication.

Mini check

Which function acts first in a composition?

Learning Objectives

Key idea

2.7.A — Evaluate the composition of two or more functions for given values. 2.7.B — Construct a representation of the composition of two or more functions. 2.7.C — Rewrite a given function as a composition of two or more functions.

Fast decision rule

Match the prompt verb to the representation it asks you to build or interpret.

Worked steps

$1$ . Name the given quantities. $2$ . Choose the matching function or representation. $3$ . Show the parameter or operation that changes. $4$ . Check one value in context.

Common mistake

Starting calculations before identifying what the input and output mean.

Mini check

Can you state what the input, output, and requested evidence are before choosing a formula?

Key Concepts

Key idea

The order of composition matters, and the domain of the composition is limited by both functions.

Fast decision rule

Draw an input-output pipeline before simplifying symbols.

Worked steps

If an inner output is not in the outer function’s domain, that input is excluded from the composition.

Common mistake

Assuming two compositions are equal because both use the same functions.

Mini check

When can two compositions have different domains?

Relevant Representations

Key idea

A composition can be represented by a pipeline, table, equation, or graph of the combined process.

Fast decision rule

Use a pipeline to preserve order and a table to track intermediate values.

Composition pipeline
Stage	Input	Output
Inner function	$x$	$g\left(x\right)$
Outer function	$g\left(x\right)$	$f\left(g\left(x\right)\right)$
Combined rule	$x$	$\left(f\circ g\right) \left(x\right)$

Worked steps

Write input $\to$ inner output $\to$ final output, then verify one input by substitution.

Common mistake

Skipping the intermediate output and losing a unit conversion or restriction.

Mini check

What belongs in the middle of a two-stage table?

Essential Relationships or Formulas

Key idea

Use an exact relationship to expose the pattern before substituting values.

(f\circ g)(x)=f(g(x))

(g\circ f)(x)=g(f(x))

Fast decision rule

Keep the variables and restrictions visible, then substitute with units attached.

Worked example

Write the relationship, identify each parameter, substitute the known values, and check one output.

Common mistake

Using a familiar formula without checking its domain, base, interval, or unit assumptions.

Mini check

Which symbol or parameter carries the feature the prompt asks you to find?

Worked Examples

Key idea

A short, labeled chain of steps is easier to check than a page of unexplained algebra.

Fast decision rule

Write the representation first, calculate second, and interpret last.

Worked steps

Place the functions in a pipeline, evaluate the inner output, then use it as the outer input.

Common mistake

Skipping the interpretation step and reporting an answer with no units, domain, or model evidence.

Mini check

Which step would you show so another student could reproduce the result?

Common Mistakes

Key idea

Most errors come from a mismatch between the pattern, the operation, and the allowed input.

Fast decision rule

Before submitting, check pattern, operation, domain, and units in that order.

Worked steps

Circle the first line where the pattern or restriction changes. Correct that line, then recompute only the affected steps.

Common mistake

Trusting a calculator output that hides a wrong model, sign, base, or domain.

Mini check

What independent check would catch your most likely error?

AP Exam Tips

Key idea

Use the topic workflow to connect the representation, the equation, and the context before calculating.

Fast decision rule

Read the prompt, choose the smallest reliable representation, and verify the result in context.

Worked steps

$1$ . Identify the pattern. $2$ . Write the matching form. $3$ . Show the key substitution or transformation. $4$ . Check units, domain, and direction.

Common mistake

Submitting a number without explaining why the selected model or transformation fits.

Mini check

Can you name the evidence that supports your final model or answer?

Practice

Open this topic’s practice set to apply the method. Work independently, then compare the evidence in your reasoning with the checked explanation.

TRY IT IN A NEW SETTING

Can you transfer the method?

For $f(x)=\sqrt{x}$ and $g(x)=x-1$ , give $f(g(x))$ and its domain.

Compare your reasoning

The composition is $\sqrt{x-1}$ . The inner output must satisfy $x-1\ge 0$ , so the domain is $x\ge 1$ .

Use it in a real question

Work independently, then compare the mathematical evidence in your answer.

Practise this topic Check the whole unit

2.7 · Composition of Functions

Topic 2.7 · Composition of Functions

What a strong response shows

What You Need to Know

Learning Objectives

Key Concepts

Relevant Representations

Essential Relationships or Formulas

Worked Examples

Common Mistakes

AP Exam Tips

Can you transfer the method?