BetterGrades Precalculus · Unit 6 · Lesson

Rational functions and their domains

Identify rational functions and determine all excluded inputs from the original denominator before simplifying.

Opening

Start with the situation

A rational function $\frac{P}{Q}$ excludes every real input where the original denominator Q is zero.

Rational graphs are organized around the inputs the denominator forbids. Those exclusions divide the graph into continuity intervals and explain why a simplified expression may still contain a hole or asymptote.

Before you begin

Prerequisite check

Factor numerator and denominator.
Preserve original restrictions.
Use sign and asymptotic notation.

Core explanation

Explanation

Factor the denominator, solve $Q=0,$ record exclusions, and partition the domain into intervals of continuity.

A rational function can have all-real domain when its denominator has no real zero.

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through numerator-denominator role map, domain intervals, or another equivalent representation.

Conceptual reading

What the idea is really doing

A rational function carries permanent memory of its original denominator. Factoring may reveal holes, asymptotes, and sign changes, but cancellation never restores an input that the original formula excluded.

This lesson narrows that lens to one goal: identify rational functions and determine all excluded inputs from the original denominator before simplifying. 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

Factor the denominator.
Solve $Q=0$ .
Record exclusions.
Partition the domain into intervals of continuity.

Verification: Record exclusions first, then compare the factored and simplified forms. Test one point in every sign interval and examine both sides of each vertical asymptote.

Foundation walkthrough

Plan before calculating

Problem

Domain of $\frac{x}{x^2-x-6}$ .

Plan: Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Factor the denominator, solve $Q=0,$ record exclusions, and partition the domain into intervals of continuity.
Conclusion: $x\ne 3,-2$
Why the check works: The denominator creates three continuity intervals.

Worked examples

See the idea in three forms

foundation example

Domain of $\frac{x}{x^2-x-6}$ .

Solution $x\ne 3,-2$

The denominator creates three continuity intervals.

representation example

Domain of $\frac{x-2}{x^2-9}$ .

Solution $x\ne \pm 3$

This example expresses rational functions and their domains in a second form.

transfer example

Do numerator zeros restrict domain?

SolutionNo.

A rational function can have all-real domain when its denominator has no real zero.

Numerator-denominator role map. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The denominator creates three continuity intervals. — **Anchor figure · Numerator-denominator role map**
Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The denominator creates three continuity intervals.

Domain intervals. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for rational functions and their domains. — **Mechanism figure · Domain intervals**
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for rational functions and their domains.

No-real-zero denominator. Compare the valid path with the tempting shortcut. The figure shows why simplifying before recording the original exclusions leads to a false conclusion. — **Comparison and error figure · No-real-zero denominator**
Compare the valid path with the tempting shortcut. The figure shows why simplifying before recording the original exclusions leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is simplifying before recording the original exclusions.

Check yourself

Domain of $\frac{1}{x+5}$ .

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 of $\frac{1}{x+5}$ .

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 of $\frac{x-2}{x^2-9}$ .

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 303

Do numerator zeros restrict 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

Can rational function have all-real domain?

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Practice 505

Explain why this conclusion is valid: $x\ne 3,-2$ . Use the foundation problem as evidence: Domain of $\frac{x}{x^2-x-6}$ .

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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 rational functions and their domains that it illustrates: Domain of $\frac{x-2}{x^2-9}$

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Practice 707

Correct this reasoning and identify the first unsafe assumption: A frequent error is simplifying before recording the original exclusions.

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Practice 808

Connect two representations for this example: Domain of $\frac{x}{x^2-x-6}$ . 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: Do numerator zeros restrict domain? 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 rational functions and their domains, then apply it to one worked example from this lesson.

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Lesson close

Connect forward

The next lesson, Equivalent formulas, different functions, and holes, 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
Stitz and Zeager, Precalculus

No long source passage is reproduced.

Domain of xx2−x−6\frac{x}{x^2-x-6}x2−x−6x​.

Domain of xx2−x−6\frac{x}{x^2-x-6}x2−x−6x​.

Domain of x−2x2−9\frac{x-2}{x^2-9}x2−9x−2​.

Do numerator zeros restrict domain?

Original BetterGrades manuscript, rights-separated references.

Domain of $\frac{x}{x^2-x-6}$ .

Domain of $\frac{x}{x^2-x-6}$ .

Domain of $\frac{x-2}{x^2-9}$ .