Introduction to Functions

⚙️ Introduction to Functions

Part 1 of 5 — Concept Introduction

Think about a vending machine. You press a button (the input), and exactly one snack drops out (the output). You would be pretty upset if you pressed B4 and sometimes got chips and sometimes got a candy bar! A good machine gives you one predictable result for each button.

That is the whole idea behind a function.

What Is a Function?

A function is a relationship where each input has exactly one output.

• The input is the value you put in (often called $x$).
• The output is the value you get back (often called $y$).

The key word is exactly one. One input can never lead to two different outputs.

A Simple Example

$y = 2x + 3$

For each value of $x$, there is exactly one value of $y$:

Input $x$	Calculation	Output $y$
$0$	$2(0) + 3$	$$

Notice that no input ever produces two different outputs. That makes this a function. ✅

Function Notation

Mathematicians have a special, tidy way to write functions called function notation:

$f(x) = 2x + 3$

You read this as "f of x equals 2x plus 3."

⚠️ Careful: $f(x)$ does not mean "f times x." The $f$ is the of the function, and the inside the parentheses is the input you are plugging in.

Concept Check 🎯

Remember the defining rule: each input must give exactly one output.

📝 Worked Examples

Part 2 of 5 — Evaluating Functions Step by Step

When you evaluate a function, you find the output for a specific input. Follow the same steps every time:

1. Write the function rule.
2. Replace every $x$ with the input value (use parentheses!).
3. Simplify using order of operations.

Example 1: Evaluate $f(x) = 3x - 5$ at $x=4$

🧭 Guided Practice

Part 3 of 5 — Guided Practice

There are four ways to represent the same function: an equation, a table, a graph, and words. Let's practice spotting and using them.

Identify the Representation 🔎

Each item below is one of the four ways to represent a function. Choose the correct name for each.

• Item A: $f(x) = 4x - 2$
• Item B: "The output is three times the input."

🌎 Functions in the Real World

Part 4 of 5 — Application & Word Problems

Functions are everywhere once you start looking. Anytime one quantity depends on another in a predictable way, you have a function.

Renting a Bike 🚲

A bike-share company charges a flat $3 fee plus $2 for every hour you ride. We can write the total cost as a function of the number of hours, $h$:

$C(h) = 2h + 3$

• The input $$ is the number of hours you ride.

🏆 Review & Challenge

Part 5 of 5 — Putting It All Together

You have learned what a function is, how to read function notation, the four ways to represent functions, and how to test whether something is a function. Here is the big picture:

Function Cheat Sheet

Idea	What to Remember
Function	Each input has exactly one output
Notation	$f(x)$ means "the output when the input is $x$"
Evaluate	Replace $x$ with the input, then simplify