BetterGrades Precalculus · Unit 2 · Lesson

Domain from formulas

Determine the real domain of a function by combining denominator, even-root, logarithm, and composition restrictions.

Opening

Start with the situation

The real domain is the intersection of every input condition imposed by denominators, even roots, logarithms, and nested functions.

The same dependency may arrive as a story, table, graph, or formula. Learning to preserve the inputs, outputs, units, and domain while moving among those forms is the central language of the course.

Before you begin

Prerequisite check

Use function notation from the algebra and function readiness unit.
Read ordered pairs and interval notation.
Attach units to contextual quantities.

Core explanation

Explanation

Write each restriction separately, solve it, intersect the solution sets, and retain exclusions hidden by cancellation.

A square root in a denominator requires a strictly positive radicand, while an ordinary square root allows zero.

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through restriction icons, condition intersection, or another equivalent representation.

Conceptual reading

What the idea is really doing

A function is a dependency, not merely an equation. Inputs, outputs, units, and domain must agree in words, tables, graphs, and formulas; each representation should tell the same mathematical story.

This lesson narrows that lens to one goal: determine the real domain of a function by combining denominator, even-root, logarithm, and composition restrictions. 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 each restriction separately.
Solve it.
Intersect the solution sets.
Retain exclusions hidden by cancellation.

Verification: Choose two representations and make them verify one another. A table can test a formula, a graph can expose a domain or range claim, and units can reveal a model that is algebraically neat but conceptually wrong.

Foundation walkthrough

Plan before calculating

Problem

Find domain of $\frac{ln(x-1)}{sqrt}(x+2)$

Plan: Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Write each restriction separately, solve it, intersect the solution sets, and retain exclusions hidden by cancellation.
Conclusion: $x>1$
Why the check works: The logarithm and denominator-root conditions are intersected.

Worked examples

See the idea in three forms

foundation example

Find domain of $\frac{ln(x-1)}{sqrt}(x+2)$

Solution $x>1$

The logarithm and denominator-root conditions are intersected.

representation example

Domain of $sqrt(2-x)$ .

Solution $x\le 2$

This example expresses domain from formulas in a second form.

transfer example

Domain of $ln(x+3)$ .

Solution $x>-3$

A square root in a denominator requires a strictly positive radicand, while an ordinary square root allows zero.

Restriction icons. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The logarithm and denominator-root conditions are intersected. — **Anchor figure · Restriction icons**
Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The logarithm and denominator-root conditions are intersected.

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

Retained hole. Compare the valid path with the tempting shortcut. The figure shows why checking only one restriction or using the simplified formula's larger domain leads to a false conclusion. — **Comparison and error figure · Retained hole**
Compare the valid path with the tempting shortcut. The figure shows why checking only one restriction or using the simplified formula's larger domain leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is checking only one restriction or using the simplified formula's larger domain.

Check yourself

Domain of $\frac{1}{x+4}$ .

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 $\frac{1}{x+4}$ .

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Attempt once to unlock the answer

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

Practice 202

Domain of $sqrt(2-x)$ .

Write a complete attempt before opening the exact answer.

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

Practice 303

Domain of $ln(x+3)$ .

Write a complete attempt before opening the exact answer.

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

Practice 404

Domain of $sqrt(x^2-9)$ .

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Practice 505

Explain why this conclusion is valid: $x>1$ . Use the foundation problem as evidence: Find domain of $\frac{ln(x-1)}{sqrt}(x+2)$ .

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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 domain from formulas that it illustrates: Domain of $sqrt(2-x)$

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Practice 707

Correct this reasoning and identify the first unsafe assumption: A frequent error is checking only one restriction or using the simplified formula's larger domain.

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Practice 808

Connect two representations for this example: Find domain of $\frac{ln(x-1)}{sqrt}(x+2)$ . 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 of $ln(x+3)$ . 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 domain from formulas, then apply it to one worked example from this lesson.

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Lesson close

Connect forward

The next lesson, Domain from context, uses this result as part of a larger structure.

Source record

Original BetterGrades manuscript, rights-separated references.

Yoshiwara, Modeling, Functions, and Graphs
Lippman and Rasmussen, Precalculus Volume 1
Stitz and Zeager, Precalculus

No long source passage is reproduced.

Find domain ofln(x−1)sqrt(x+2)\frac{ln(x-1)}{sqrt}(x+2)sqrtln(x−1)​(x+2)

Find domain ofln(x−1)sqrt(x+2)\frac{ln(x-1)}{sqrt}(x+2)sqrtln(x−1)​(x+2)

Domain of sqrt(2−x)sqrt(2-x)sqrt(2−x).

Domain of ln(x+3)ln(x+3)ln(x+3).

Original BetterGrades manuscript, rights-separated references.

Find domain of $\frac{ln(x-1)}{sqrt}(x+2)$

Find domain of $\frac{ln(x-1)}{sqrt}(x+2)$

Domain of $sqrt(2-x)$ .

Domain of $ln(x+3)$ .