BetterGrades Precalculus · Unit 6 · Lesson

Equivalent formulas, different functions, and holes

Identify removable discontinuities caused by common factors and distinguish formula equivalence from function equality.

Opening

Start with the situation

A common factor creates a removable discontinuity: the original function follows the simplified curve but omits one point.

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, retain original exclusions, cancel, and evaluate the simplified formula at the cancelled input to locate the hole.

Two formulas define the same function only when both domains and values match.

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through cancellation with domain memory, hole coordinate construction, 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 removable discontinuities caused by common factors and distinguish formula equivalence from function equality. 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.
Retain original exclusions.
Cancel.
Evaluate the simplified formula at the cancelled input to locate the hole.

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

Analyze $\frac{x^2-9}{x-3}$ .

Plan: Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Factor, retain original exclusions, cancel, and evaluate the simplified formula at the cancelled input to locate the hole.
Conclusion: Line $y=x+3$ with hole $(3,6)$ .
Why the check works: The original denominator excludes $3$ .

Worked examples

See the idea in three forms

foundation example

Analyze $\frac{x^2-9}{x-3}$ .

SolutionLine $y=x+3$ with hole $(3,6)$ .

The original denominator excludes $3$ .

representation example

Hole of $\frac{x^2+3x+2}{x+1}$ .

Solution $(-1,1)$

This example expresses equivalent formulas, different functions, and holes in a second form.

transfer example

Can a function have two holes?

SolutionYes.

Two formulas define the same function only when both domains and values match.

Cancellation with domain memory. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The original denominator excludes 3. — **Anchor figure · Cancellation with domain memory**
Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The original denominator excludes $3$ .

Hole coordinate construction. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for equivalent formulas, different functions, and holes. — **Mechanism figure · Hole coordinate construction**
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for equivalent formulas, different functions, and holes.

Equal expressions versus functions. Compare the valid path with the tempting shortcut. The figure shows why filling the hole because the simplified formula has a value there leads to a false conclusion. — **Comparison and error figure · Equal expressions versus functions**
Compare the valid path with the tempting shortcut. The figure shows why filling the hole because the simplified formula has a value there leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is filling the hole because the simplified formula has a value there.

Check yourself

Hole of $\frac{x^2-1}{x-1}$ .

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

Hole of $\frac{x^2-1}{x-1}$ .

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

Hole of $\frac{x^2+3x+2}{x+1}$ .

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

Can a function have two holes?

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

What sets hole y-value?

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: Line $y=x+3$ with hole $(3,6)$ . Use the foundation problem as evidence: Analyze $\frac{x^2-9}{x-3}$ .

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 equivalent formulas, different functions, and holes that it illustrates: Hole of $\frac{x^2+3x+2}{x+1}$

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 filling the hole because the simplified formula has a value there.

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: Analyze $\frac{x^2-9}{x-3}$ . 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: Can a function have two holes? 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 equivalent formulas, different functions, and holes, 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, Vertical asymptotes and local behavior, 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.

Analyze x2−9x−3\frac{x^2-9}{x-3}x−3x2−9​.

Analyze x2−9x−3\frac{x^2-9}{x-3}x−3x2−9​.

Hole of x2+3x+2x+1\frac{x^2+3x+2}{x+1}x+1x2+3x+2​.

Can a function have two holes?

Original BetterGrades manuscript, rights-separated references.

Analyze $\frac{x^2-9}{x-3}$ .

Analyze $\frac{x^2-9}{x-3}$ .

Hole of $\frac{x^2+3x+2}{x+1}$ .