BetterGrades Precalculus · Unit 4 · Lesson

Domains of composite functions

Determine the domain of a composite by enforcing both inner-function and outer-input conditions.

Opening

Start with the situation

The domain of f(g(x)) requires $x$ to be allowed by $g$ and $g(x)$ to be allowed by $f$ .

Many real models are built in stages—a conversion followed by a cost rule, or a measurement followed by a calibration. Composition records that order, while inverse reasoning asks whether the stages can be undone.

Before you begin

Prerequisite check

Evaluate function notation.
Determine domains from formulas.
Solve equations for a selected variable.

Core explanation

Explanation

Translate outer restrictions into equations or inequalities involving the inner output, then intersect them with the inner domain.

One forbidden outer input can have several original preimages.

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through composite-domain pipeline, preimage of forbidden value, or another equivalent representation.

Conceptual reading

What the idea is really doing

Composition and inversion are about information flow. A composite sends an input through stages in a fixed order; an inverse reverses that flow only when each output identifies a unique input.

This lesson narrows that lens to one goal: determine the domain of a composite by enforcing both inner-function and outer-input conditions. The point is not to memorize an isolated trick; it is to know what evidence makes the conclusion valid and how a second representation can check it.

Reusable method

A reliable route through the problem

Translate outer restrictions into equations or inequalities involving the inner output.
Then intersect them with the inner domain.
Can the process be reversed without ambiguity?.

Verification: Name the intermediate quantity, enforce its domain, and verify an inverse with composition. Units are especially useful because the output unit of one stage must match the input unit of the next.

Foundation walkthrough

Plan before calculating

Problem

$f=sqrt(u),g=x^2-9$

Plan: Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Translate outer restrictions into equations or inequalities involving the inner output, then intersect them with the inner domain.
Conclusion: Domain $x\le -3$ or $x\ge 3$ .
Why the check works: The outer root condition becomes $x^2-9\ge 0$ .

Worked examples

See the idea in three forms

foundation example

$f=sqrt(u),g=x^2-9$

SolutionDomain $x\le -3$ or $x\ge 3$ .

The outer root condition becomes $x^2-9\ge 0$ .

representation example

Domain $\frac{1}{x^2-16}$ .

Solution $x\ne \pm 4$

This example expresses domains of composite functions in a second form.

transfer example

Domain $ln(4-x^2)$ .

Solution $(-2,2)$

One forbidden outer input can have several original preimages.

Composite-domain pipeline. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The outer root condition becomes x^2-9≥0. — **Anchor figure · Composite-domain pipeline**
Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The outer root condition becomes $x^2-9\ge 0$ .

Preimage of forbidden value. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for domains of composite functions. — **Mechanism figure · Preimage of forbidden value**
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for domains of composite functions.

Hidden restriction. Compare the valid path with the tempting shortcut. The figure shows why simply intersecting the written domains of f and g as though they used the same stage variable leads to a false conclusion. — **Comparison and error figure · Hidden restriction**
Compare the valid path with the tempting shortcut. The figure shows why simply intersecting the written domains of $f$ and $g$ as though they used the same stage variable leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is simply intersecting the written domains of $f$ and $g$ as though they used the same stage variable.

Check yourself

Domain $sqrt(x-5)$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice

Ten concrete questions

Practice 101

Domain $sqrt(x-5)$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 202

Domain $\frac{1}{x^2-16}$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 303

Domain $ln(4-x^2)$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 404

Two domain questions.

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Practice 505

Explain why this conclusion is valid: Domain $x\le -3$ or $x\ge 3$ . Use the foundation problem as evidence: $f=sqrt(u),g=x^2-9$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 606

Solve the representation example, then name the feature of domains of composite functions that it illustrates: Domain $\frac{1}{x^2-16}$

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Practice 707

Correct this reasoning and identify the first unsafe assumption: A frequent error is simply intersecting the written domains of $f$ and $g$ as though they used the same stage variable.

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Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 808

Connect two representations for this example: $f=sqrt(u),g=x^2-9$ . Describe what a graph, table, mapping, or algebraic form would have to show.

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Practice 909

Create a nearby example by changing one number or condition in this prompt: Domain $ln(4-x^2)$ . Predict the effect, solve your new example, and compare it with the original.

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Practice 1010

Write a short verification checklist for domains of composite functions, then apply it to one worked example from this lesson.

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Lesson close

Connect forward

The next lesson, One-to-one functions, uses this result as part of a larger structure.

Source record

Original BetterGrades manuscript, rights-separated references.

Lippman and Rasmussen, Precalculus Volume 1
Utah College Algebra
AP Precalculus framework

No long source passage is reproduced.

f=sqrt(u),g=x2−9f=sqrt(u),g=x^2-9f=sqrt(u),g=x2−9

f=sqrt(u),g=x2−9f=sqrt(u),g=x^2-9f=sqrt(u),g=x2−9

Domain 1x2−16\frac{1}{x^2-16}x2−161​.

Domain ln(4−x2)ln(4-x^2)ln(4−x2).

Original BetterGrades manuscript, rights-separated references.

$f=sqrt(u),g=x^2-9$

$f=sqrt(u),g=x^2-9$

Domain $\frac{1}{x^2-16}$ .

Domain $ln(4-x^2)$ .