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Mathematics

Introduction to Functions

A High School & Early College Primer

Functions show up on every algebra quiz, every precalculus exam, and every standardized test — and most students hit a wall the moment the notation switches from y = to f(x) =. If you've stared at a problem involving domain and range or composite functions and felt completely lost, this guide is written for you.

**TLDR: Introduction to Functions** is a focused, 10–20 page primer that covers exactly what a high school or early college student needs: what a function actually is (and how to spot one), how to read and use function notation, how to find domain and range without guessing, the six function families you'll see repeatedly in algebra and precalculus, and how transformations shift and stretch a graph in predictable ways. The final section walks through combining functions and understanding inverses — including when an inverse exists and how to find it.

This is not a 400-page textbook. Every explanation is direct, every term is defined the first time it appears, and every concept comes with a worked example. It's the right length to read the night before a test or alongside a confusing chapter in your current course. Parents and tutors prepping for a session will find it just as useful.

If you're looking for a no-nonsense algebra 2 functions quick review, pick this up and be ready for class.

What you'll learn
  • Define a function and test whether a relation is one using the vertical line test and the input-output rule
  • Read and evaluate function notation, including domain and range
  • Recognize the graphs and behavior of linear, quadratic, absolute value, square root, exponential, and piecewise functions
  • Apply transformations (shifts, stretches, reflections) to a parent function
  • Combine functions through arithmetic and composition, and find inverses when they exist
What's inside
  1. 1. What Is a Function?
    Defines a function as a rule that assigns exactly one output to each input, with the vertical line test and everyday analogies.
  2. 2. Function Notation, Domain, and Range
    Introduces f(x) notation, evaluating functions at specific values, and how to find the set of allowed inputs and possible outputs.
  3. 3. The Function Families You'll See Everywhere
    Surveys linear, quadratic, absolute value, square root, exponential, and piecewise functions with their graphs and key features.
  4. 4. Transformations: Shifting, Stretching, and Flipping
    Shows how changes inside and outside the function move and reshape its graph in predictable ways.
  5. 5. Combining Functions and Inverses
    Covers adding, multiplying, and composing functions, plus what an inverse function is and when one exists.
Published by Solid State Press
Introduction to Functions cover
TLDR STUDY GUIDES

Introduction to Functions

A High School & Early College Primer
Solid State Press

Introduction to Functions

A High School & Early College Primer

Copyright © 2026 Solid State Press.

Published by Solid State Press.

All rights reserved. No part of this book may be reproduced or transmitted in any form without prior written permission, except for brief quotations in critical reviews and educational use.

Part of the TLDR Study Guides series.

Who This Book Is For

If you are a high school student who needs a clear what-is-a-function math explainer before an Algebra 2 test, a precalculus student who keeps losing points on domain and range, or a parent helping a kid with algebra functions for the first time, this book is for you. It also works for community college students in a developmental math course and for anyone who wants a fast reset before a placement exam.

This intro to functions algebra study guide covers five tightly focused topics: what a function is and how to test for one, function notation and high school math help on reading $f(x)$ expressions, domain and range practice for beginners, the core function families from linear to exponential, transformations, and combining or inverting functions. Think of it as an algebra 2 functions quick review book that doubles as a precalculus functions primer for students who need the bigger picture. About 15 pages, no filler.

Read straight through once, then go back and work every example by hand before checking the solution. Finish with the problem set at the end to confirm what stuck.

Contents

  1. 1 What Is a Function?
  2. 2 Function Notation, Domain, and Range
  3. 3 The Function Families You'll See Everywhere
  4. 4 Transformations: Shifting, Stretching, and Flipping
  5. 5 Combining Functions and Inverses
Chapter 1

What Is a Function?

A function is a rule that assigns exactly one output to each input. That single sentence is the whole idea — everything else is unpacking it.

Start with something familiar. When you type a number into a calculator and press the square-root key, you get back exactly one number. Type 25, get 5. Type 9, get 3. The calculator never pauses and returns two answers, and it never shrugs and returns nothing (for non-negative inputs, at least). That calculator key is behaving like a function: one input in, one output out, every time.

Inputs, Outputs, and Relations

The value you feed into a function is called the input (also called the argument, though you'll hear "input" more often at this level). The value the function produces is the output. A function is a specific kind of relation — a pairing between inputs and outputs. What makes a function special is the exactly one output requirement.

Not every relation is a function. A relation is just any set of input-output pairs; a function is a relation where no input is paired with more than one output. Think of it as a rule at a very strict club: each member (input) may bring exactly one guest (output). An input can sit alone — some inputs simply have no output, and we'll talk about that when we cover domain in the next section. But an input can never bring two guests. That's the rule that separates functions from relations in general.

Keep reading

You've read the first half of Chapter 1. The complete book covers 5 chapters in roughly fifteen pages — readable in one sitting.

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