BetterGrades Precalculus · Unit 4 · Lesson

Composition from formulas

Evaluate and simplify composite functions symbolically while preserving grouping and domain conditions.

Opening

Start with the situation

Formula composition replaces every occurrence of the outer variable with the complete inner expression.

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

Write the substitution with parentheses before simplifying and determine both inner and outer domain conditions.

Simplification may hide restrictions inherited from an intermediate stage.

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through substitution slots, expression trees, 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: evaluate and simplify composite functions symbolically while preserving grouping and domain 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

Write the substitution with parentheses before simplifying.
Determine both inner.
Outer domain conditions.

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

$f=x^2-1,g=3x+2$

Plan: Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Write the substitution with parentheses before simplifying and determine both inner and outer domain conditions.
Conclusion: $f(g(x))=9x^2+12x+3$
Why the check works: The entire linear expression is squared.

Worked examples

See the idea in three forms

foundation example

$f=x^2-1,g=3x+2$

Solution $f(g(x))=9x^2+12x+3$

The entire linear expression is squared.

representation example

$g$ after $f$ .

Solution $(2x+3)^2$

This example expresses composition from formulas in a second form.

transfer example

$f=\frac{1}{x},g=x-4$ ; domain.

Solution $x\ne 4$

Simplification may hide restrictions inherited from an intermediate stage.

Substitution slots. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The entire linear expression is squared. — **Anchor figure · Substitution slots**
Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The entire linear expression is squared.

Expression trees. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for composition from formulas. — **Mechanism figure · Expression trees**
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for composition from formulas.

Outer restriction preimage. Compare the valid path with the tempting shortcut. The figure shows why substituting into only one term or dropping grouping symbols leads to a false conclusion. — **Comparison and error figure · Outer restriction preimage**
Compare the valid path with the tempting shortcut. The figure shows why substituting into only one term or dropping grouping symbols leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is substituting into only one term or dropping grouping symbols.

Check yourself

$f=2x+3,g=x^2$ ; $f$ after $g$ .

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

$f=2x+3,g=x^2$ ; $f$ after $g$ .

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

$g$ after $f$ .

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

$f=\frac{1}{x},g=x-4$ ; domain.

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

Why parentheses?

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 505

Explain why this conclusion is valid: $f(g(x))=9x^2+12x+3$ . Use the foundation problem as evidence: $f=x^2-1,g=3x+2$ .

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 606

Solve the representation example, then name the feature of composition from formulas that it illustrates: $g$ after $f$ .

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 707

Correct this reasoning and identify the first unsafe assumption: A frequent error is substituting into only one term or dropping grouping symbols.

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 808

Connect two representations for this example: $f=x^2-1,g=3x+2$ . Describe what a graph, table, mapping, or algebraic form would have to show.

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 909

Create a nearby example by changing one number or condition in this prompt: $f=\frac{1}{x},g=x-4$ ; domain. Predict the effect, solve your new example, and compare it with the original.

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 1010

Write a short verification checklist for composition from formulas, then apply it to one worked example from this lesson.

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.

Lesson close

Connect forward

The next lesson, Composition from tables and graphs, 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=x2−1,g=3x+2f=x^2-1,g=3x+2f=x2−1,g=3x+2

f=x2−1,g=3x+2f=x^2-1,g=3x+2f=x2−1,g=3x+2

ggg after fff.

f=1x,g=x−4f=\frac{1}{x},g=x-4f=x1​,g=x−4; domain.

Original BetterGrades manuscript, rights-separated references.

$f=x^2-1,g=3x+2$

$f=x^2-1,g=3x+2$

$g$ after $f$ .

$f=\frac{1}{x},g=x-4$ ; domain.