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Math workbook

College Algebra: Functions and Graphs: Domains, composition, inverses, and models

Intro CollegeLevel 5
Cover of College Algebra: Functions and Graphs: Domains, composition, inverses, and models

About this book

Read every function closely: its inputs, its outputs, its restrictions.

This workbook gives introductory college learners focused practice with functions. It works through four connected skills: finding and justifying domains, transforming and composing functions, building inverses on a suitable domain, and comparing polynomial, rational, exponential and logarithmic models. Each module explains one idea plainly, shows worked examples, then fades support from guided to independent practice, error analysis and spaced review. An explanatory answer key names common errors and gives a reteaching cue for every page. It is a practice companion, not a full course.

  • A readiness check with a clear starting route
  • Four modules with worked examples and faded practice
  • Error analysis built on common misconceptions
  • Cumulative review, exit check and parallel retry
  • Answer key with reasons and reteaching cues

For college and advanced high school learners who can graph lines and factor, solve and graph quadratics.

Look inside

Page 5 of College Algebra: Functions and Graphs. Step 2, Concept: Functions, domains, and representations
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Page 6 of College Algebra: Functions and Graphs. Step 3, Worked Example: Functions, domains, and representations
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What this book covers

Covered

  • Target skill. You will read functions across formulas, tables and graphs, justify domains, compose and invert functions, and compare polynomial, rational, exponential and logarithmic models.
  • In scope: function notation, natural and context domains, interval notation, piecewise rules, transformations, completing the square, composition, inverses on restricted domains, intercepts, asymptotes and holes, rational equations, growth and decay rates, and basic exponential and logarithmic equations.

Not covered

Out of scope: trigonometric functions, systems, matrices, sequences, and formal limits.

Before you start

Before and after. You should already solve, factor and graph linear and quadratic equations (see Linear Equations and Graphs and Quadratics: Patterns, Graphs, and Solutions).

Things to have

Use a graphing tool to check, but decide domains from the algebra and the situation.

Notice

Pictures in this book are simplified learning models. Check answers with the explanations at the back of the book.

The steps in this book

How the steps work
  1. Step 1: Readiness Check

    Check the skills you need before you begin. If something feels tricky, an adult can help you start in the right place.

  2. Step 2: Concept

    Read a short, clear explanation of the new idea and the words that go with it.

  3. Step 3: Worked Example

    Watch the idea being used, one step at a time, in a worked example.

  4. Step 4: Guided Practice

    Practice with prompts and help cues. The help gets smaller as you get stronger.

  5. Step 5: Independent Practice

    Practice on your own with different kinds of questions, and explain your thinking.

  6. Step 6: Review

    Look back at earlier learning so it stays strong, then check where to go next.

  7. Step 7: Answers and Explanations

    Check your answers. Each answer explains why, names a common mistake, and shows how to fix it.

What is in the book

The modules

  1. Module 1: Functions, domains, and representationsPages 5–14
  2. Module 2: Transform and combine functionsPages 15–24
  3. Module 3: Invert with appropriate restrictionsPages 25–34
  4. Module 4: Compare polynomial, rational, and growth modelsPages 35–44

Contents

  1. Readiness check, pp. 3–4. Before Module 1. Its table tells you where to start.
  2. Modules, pp. 5–44
  3. Checkpoints, pp. 14, 24, 34, 44. The last page of each module.
  4. Mixed review, pp. 45–48. Questions from across the book, mixed together.
  5. Exit check, pp. 49–50. A last check across the book.
  6. Reflect and repair, p. 51
  7. Try again, p. 52. Fresh questions like those in the exit check.
  8. Further practice, pp. 53–56. More pages of step 5, Independent Practice.
  9. Extra practice, pp. 57–58
  10. Answers and explanations, pp. 59–68. Notes in this part are headed “Why”, “Common mistake”, “Help” and “Look for”.

Before this book

You should already solve, factor and graph linear and quadratic equations (see Linear Equations and Graphs and Quadratics: Patterns, Graphs, and Solutions).

After this book

A natural next book is Calculus: Limits and First Derivatives.

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