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 $y$
$0$	$2(0) + 3$	$3$
$1$	$2(1) + 3$	$5$
$2$	$2(2) + 3$	$7$

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 name of the function, and the $x$ inside the parentheses is the input you are plugging in.

Function notation is handy because it tells you the input right there in the parentheses. When you see $f(4)$, it is asking: "What is the output when the input is 4?"

Finding $f(4)$

To find $f(4)$, replace every $x$ with $4$:

$f(4) = 2(4) + 3 = 8 + 3 = 11$

So $f(4) = 11$. The input $4$ produces the output $11$.

Quick practice in your head: What is $f(0)$? Replace $x$ with $0$: $f(0) = 2(0) + 3 = 3$.

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 $x = 4$

• Write: $f(x) = 3x - 5$
• Replace $x$ with $4$: $f(4) = 3(4) - 5$
• Simplify: $f(4) = 12 - 5 = 7$

So $f(4) = 7$. ✅

Example 2: Evaluate $g(x) = x^2 + 1$ at $x = 3$

• Write: $g(x) = x^2 + 1$
• Replace $x$ with $3$: $g(3) = (3)^2 + 1$
• Simplify: $g(3) = 9 + 1 = 10$

So $g(3) = 10$. The parentheses keep you from making sign and order mistakes.

💡 Tip: Always put the input inside parentheses when you substitute. This matters a lot when the input is negative, like $f(-2)$.

Your Turn 🧮

Use the function $f(x) = 2x + 1$ to fill in the table. Replace $x$ with each input and simplify.

1. What is $f(0)$? (Type a single number.)

2. What is $f(3)$? (Type a single number.)

3. What is $f(5)$? (Type a single number.)

🧭 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 $h$ is the number of hours you ride.
• The output $C(h)$ is the total cost in dollars.

Hours $h$	Cost $C(h) = 2h + 3$	Total
$1$	$2(1) + 3$	$5
$2$	$2(2) + 3$	$7
$4$	$2(4) + 3$	$11

This is a function because each number of hours produces exactly one total cost. You will never be charged two different prices for the same ride length.

Apply It 🧮

Use the bike-rental function $C(h) = 2h + 3$, where $h$ is the number of hours and $C(h)$ is the total cost in dollars.

1. What is the total cost for a $3$-hour ride? (Type the number of dollars only.)

2. What is the total cost for a $5$-hour ride? (Type the number of dollars only.)

3. What is the cost if you ride for $0$ hours (just the flat fee)? (Type the number of dollars only.)

Application Check 🎯

Think about what the input and output mean in the real-world story.

🏆 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
Four representations	Equation, table, graph, words
Vertical Line Test	If a vertical line hits a graph more than once, it is not a function

The Vertical Line Test 📏

To check a graph, imagine sliding a vertical line across it:

• Hits the graph at one point everywhere → it is a function. ✅
• Hits the graph at two or more points anywhere → it is not a function. ❌

This works because a single input ($x$) should never line up with two different outputs ($y$).

Challenge Round 🎯

These questions mix everything from this lesson. Take your time and reason through each one.