Exponential Domain and Range

Q: How can I check my work?

At x=0 the value is negative; positivity of 2^x proves the entire range statement. A numerical check can detect an error; when claiming an identity for all inputs, also explain the algebraic property that makes it true.

Q: Which mistakes should I watch for?

Using the percent rate itself as the growth factor. Adding the same amount each period in an exponential model. Including zero in the range of an unshifted nonzero exponential.

Q: Can I use another valid method?

Yes. Explain your assumptions, show enough intermediate work to justify the result, and verify it in the original problem. Equivalent expressions and valid alternative methods receive credit.

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By Cambridge Veritas · Algebra I learning resources

Introduction

Exponential Domain and Range helps you connect a mathematical rule to a result you can explain and check. Distinguish the all-real mathematical domain from contextual time restrictions. Explain why zero is not reached by a nonzero unshifted exponential. This lesson belongs to exponential functions and builds toward independent Algebra I problem solving.

A.9A · Supporting

Alignment uses 19 TAC §111.39(c). Review lessons sample their listed standards. Readiness and supporting labels describe STAAR emphasis, not whether a topic is required.

Learning objectives

Identify the quantities and representations used in exponential domain and range.
Apply the lesson's rule to solve a numerical or contextual problem.
Explain each mathematical step using the correct vocabulary.
Verify an answer and correct a likely error.

Prerequisite knowledge

Evaluate powers and convert percentages to decimals.

Previous lesson: Comparing Linear and Exponential Models

Key mathematical vocabulary

constant: A value fixed within a problem.
domain: The set of permitted inputs.
range: The set of outputs produced.
exponent: A number specifying a power.
factor: An expression multiplied by another to form a product.

Formulas and rules

f(x)=ab^x, b>0, b≠1
Growth factor=1+r; decay factor=1−r
A=P(1+r/n)^nt

Exponential growth and decay

Distinguish the all-real mathematical domain from contextual time restrictions. Explain why zero is not reached by a nonzero unshifted exponential.

An exponential model has the form f(t)=ab^t, where b>0 and b≠1 for a nonconstant exponential. The initial value is a=f(0). Each increase of 1 in t multiplies the output by b rather than adding a constant.

For a positive initial amount, b>1 models growth and 0<b<1 models decay. A rate r written as a decimal gives growth factor 1+r or decay factor 1−r. Compound interest is A=P(1+r/n)^nt, with compatible units. For a≠0, the mathematical domain is all real numbers; the sign of a determines whether the range is y>0 or y<0. Context may restrict time to t≥0.

A method you can explain

Read the question and identify the known information, unknown quantities, and any domain restrictions. Select a suitable representation and tool.
Apply the relationship explained above. The positive base 2 allows every real exponent: −∞<x<∞.
Write each calculation or logical inference on a separate line. Keep units and signs visible. When two methods are possible, compare which one uses the given information most directly.
Check the result in the ORIGINAL conditions, not merely the last simplified line. At x=0 the value is negative; positivity of 2^x proves the entire range statement.
Interpret what the answer means. Decide whether its size, sign, and precision make sense; identify a restriction or assumption that would change the conclusion.

Five worked examples

Example 1 · Beginner

Find mathematical domain and range for f(x)=−350·2^x.

y = 350(2)^x. Axes may use different scales.

The positive base 2 allows every real exponent: −∞<x<∞.
For every real x, 2^x>0; multiplying by a negative number makes outputs negative.
Outputs approach 0 without reaching it and extend downward without bound, so y<0. If a=0 in ab^x, the range instead is {0}.

Answer: All real x; y < 0

Verify: At x=0 the value is negative; positivity of 2^x proves the entire range statement.

Avoid this mistake: Using the percent rate itself as the growth factor.

Example 2 · Basic

Find mathematical domain and range for f(x)=−360·2^x.

y = 360(2)^x. Axes may use different scales.

The positive base 2 allows every real exponent: −∞<x<∞.
For every real x, 2^x>0; multiplying by a negative number makes outputs negative.
Outputs approach 0 without reaching it and extend downward without bound, so y<0. If a=0 in ab^x, the range instead is {0}.

Answer: All real x; y < 0

Verify: At x=0 the value is negative; positivity of 2^x proves the entire range statement.

Avoid this mistake: Adding the same amount each period in an exponential model.

Example 3 · Intermediate

Find mathematical domain and range for f(x)=−370·2^x.

y = 370(2)^x. Axes may use different scales.

The positive base 2 allows every real exponent: −∞<x<∞.
For every real x, 2^x>0; multiplying by a negative number makes outputs negative.
Outputs approach 0 without reaching it and extend downward without bound, so y<0. If a=0 in ab^x, the range instead is {0}.

Answer: All real x; y < 0

Verify: At x=0 the value is negative; positivity of 2^x proves the entire range statement.

Avoid this mistake: Including zero in the range of an unshifted nonzero exponential.

Example 4 · Advanced

Find mathematical domain and range for f(x)=−260·2^x.

y = 260(2)^x. Axes may use different scales.

The positive base 2 allows every real exponent: −∞<x<∞.
For every real x, 2^x>0; multiplying by a negative number makes outputs negative.
Outputs approach 0 without reaching it and extend downward without bound, so y<0. If a=0 in ab^x, the range instead is {0}.

Answer: All real x; y < 0

Verify: At x=0 the value is negative; positivity of 2^x proves the entire range statement.

Avoid this mistake: Using the percent rate itself as the growth factor.

Example 5 · Challenge

Find mathematical domain and range for f(x)=−270·2^x.

y = 270(2)^x. Axes may use different scales.

The positive base 2 allows every real exponent: −∞<x<∞.
For every real x, 2^x>0; multiplying by a negative number makes outputs negative.
Outputs approach 0 without reaching it and extend downward without bound, so y<0. If a=0 in ab^x, the range instead is {0}.

Answer: All real x; y < 0

Verify: At x=0 the value is negative; positivity of 2^x proves the entire range statement.

Avoid this mistake: Adding the same amount each period in an exponential model.

Common mistakes and corrections

Using the percent rate itself as the growth factor.
Adding the same amount each period in an exponential model.
Including zero in the range of an unshifted nonzero exponential.

Real-world applications

Model a population with a constant percent change. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Predict a repeated depreciation or compound-interest balance. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.

Topic summary

Distinguish the all-real mathematical domain from contextual time restrictions. Explain why zero is not reached by a nonzero unshifted exponential.

Frequently asked questions

What should I know before exponential domain and range?

Evaluate powers and convert percentages to decimals.

How can I check my work?

At x=0 the value is negative; positivity of 2^x proves the entire range statement. A numerical check can detect an error; when claiming an identity for all inputs, also explain the algebraic property that makes it true.

Which mistakes should I watch for?

Using the percent rate itself as the growth factor. Adding the same amount each period in an exponential model. Including zero in the range of an unshifted nonzero exponential.

Can I use another valid method?

Yes. Explain your assumptions, show enough intermediate work to justify the result, and verify it in the original problem. Equivalent expressions and valid alternative methods receive credit.

Mathematical process practice · A.1A–A.1G

A.1A: Model a population with a constant percent change.
A.1B: Plan, solve, check, and interpret the written response.
A.1C: Select paper, a calculator, a table, or a graph; explain the choice.
A.1D: Connect symbols with a verbal explanation and another representation.
A.1E: Organize the solution with a table, diagram, or labeled equations.
A.1F: Explain how the representation and the algebraic rule express the same relationship.
A.1G: Justify each operation and the reasonableness of the conclusion.

Quick revision notes

Distinguish the all-real mathematical domain from contextual time restrictions. Explain why zero is not reached by a nonzero unshifted exponential.
Using the percent rate itself as the growth factor.
Adding the same amount each period in an exponential model.
Including zero in the range of an unshifted nonzero exponential.

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Original Cambridge Veritas practice. These are not official STAAR items. Cambridge Veritas is not affiliated with or endorsed by TEA.

Official Algebra I TEKS (§111.39) · TEA assessed curriculum · STAAR blueprint · TEA instructional materials

Introduction

Learning objectives

Prerequisite knowledge

Key mathematical vocabulary

Formulas and rules

Exponential growth and decay

A method you can explain

Five worked examples

Example 1 · Beginner

Example 2 · Basic

Example 3 · Intermediate

Example 4 · Advanced

Example 5 · Challenge

Common mistakes and corrections

Real-world applications

Topic summary

Frequently asked questions

Quick revision notes

Practice and explain

Part A: Four objective questions

Part B: Explain your reasoning

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