Chapter 2: Functions

Chapter 2 builds the central concept of all precalculus and calculus: the function. A function is a rule that assigns exactly one output to each input. You'll learn to recognize, graph, transform, combine, and invert functions — skills used in every chapter that follows.

Textbook alignment

📘Stewart: ~Ch 2

📗Blitzer: ~Ch 1

📙Sullivan: ~Ch 2

📕Larson: ~Ch 1

📓OpenStax: ~Ch 1

Sections

2.1Functions

A function is a rule: one input, exactly one output. The definition seems simple, but it underpins everything in higher math. This section covers what a function is, function notation, domain and range, and how to evaluate functions.

FunctionDomainRangeFunction notation+1 more

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3 concepts1 worked example9 practice problems

2.2Graphs of Functions

A function's graph is the set of all points (x, f(x)). The shape of the graph reveals information about the function's behavior — where it's increasing, decreasing, at a maximum or minimum, and whether it has any symmetry.

IncreasingDecreasingLocal maximumLocal minimum+2 more

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2 concepts1 worked example8 practice problems

2.3Getting Information from the Graph of a Function

Graphs communicate more than equations. This section teaches you to extract information directly from a graph: values, domain, range, increasing/decreasing intervals, and solutions to f(x) = c and f(x) > 0.

x-intercepty-interceptSolving f(x) = c graphicallySolving f(x) > 0 graphically

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2 concepts1 worked example7 practice problems

2.4Average Rate of Change of a Function

Average rate of change measures how fast a function changes between two points. It's the slope of the secant line. This concept is the foundation of the derivative in calculus.

Average rate of changeSecant lineDifference quotient

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1 concept1 worked example7 practice problems

2.5Linear Functions and Models

Linear functions are the simplest functions — their graphs are straight lines. The key feature is a constant rate of change. This section connects slope-intercept form to real-world modeling.

Linear functionSlopey-interceptSlope-intercept form

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2 concepts1 worked example7 practice problems

2.6Transformations of Functions

Any function can be shifted, stretched, compressed, or reflected using simple rules. Mastering transformations means you can graph complex functions just by starting with a basic parent function and applying these rules.

Vertical shiftHorizontal shiftVertical stretch/compressionReflection across x-axis+2 more

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3 concepts1 worked example8 practice problems

2.7Combining Functions

Functions can be combined by adding, subtracting, multiplying, dividing, or composing them. Composition is the most important: applying one function to the output of another. Decomposing a composite function into parts is a key calculus skill (chain rule).

Sum of functionsProduct of functionsQuotient of functionsComposition

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2 concepts1 worked example7 practice problems

2.8One-to-One Functions and Their Inverses

A one-to-one function has the property that different inputs give different outputs — each output is used at most once. These are the functions that have inverses. The inverse function undoes what the original function does.

One-to-one functionHorizontal Line TestInverse functionGraph of inverse

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2 concepts1 worked example8 practice problems

What's included — free

✓Visual concept explanations with diagrams for every section
✓Step-by-step worked examples you can study at your pace
✓Key vocabulary and memory aids for each topic
✓Printable worksheets generated for each section

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← Ch 1: Fundamentals Ch 3: Polynomial and Rational Functions →