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Introduction to Functions

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Introduction to Functions - Complete Interactive Lesson

Part 1: What Is a Function?

⚙️ Introduction to Functions

Part 1 of 5 — What Is a Function?

Topics in This Part

Section
Relations vs. Functions
The Vending-Machine Idea
The "One Output" Rule

🔑 Key Concept: A function is a rule that takes each input and gives back exactly one output. That single restriction — one output per input — is the whole idea behind everything in this lesson.

Relations vs. Functions

A relation is any set of ordered pairs $(x, y)$ — it just pairs inputs with outputs.

A function is a special relation: every input is paired with exactly one output.

Think of a vending machine. You press a button (the input) and you get one specific snack (the output). Press B4 and you always get the same chips. If pressing B4 sometimes gave chips and sometimes gave a candy bar, the machine would be broken — and that broken machine is not a function.

Input $x$	→	Output $y$
$1$	→	$3$
$2$	→

Every input above leads to one output, so this relation is a function.

🔑 The Rule: No input is allowed to have two outputs. (An output can be repeated — that's fine — but an input can never split.)

Concept Check 🎯

Spotting a Non-Function

A relation fails to be a function the moment one input points to two different outputs.

This IS a function — each input has one arrow out:

$1 \to 3, \quad 2 \to 6, \quad 3 \to 9$

This is NOT a function — the input $2$ points to two outputs:

Function or Not? 🔽

For each relation, decide whether it is a function.

A Quick Way to Test

To decide if a relation is a function, scan the inputs:

List every input (first coordinate).
If the same input ever shows up with two different outputs, it is not a function.
Otherwise, it is a function.

That is the only check you ever need for a set of ordered pairs.

Count the Outputs 🧮

A relation is a function when no input has more than one output.

1) In $\{(1, 2), (2, 4), (3, 2)\}$ , how many different inputs are there? (Answer with a number.) 2) In that same set, does any single input appear with two different outputs? (Answer 0 for no, 1 for yes.)

What You Have So Far

You can now tell a function from a plain relation: a function never lets one input split into two outputs.

In Part 2, we give the inputs and outputs their proper names — domain and range — and learn to read functions from tables, mappings, and graphs.

Part 2: Domain, Range & the Vertical Line Test

⚙️ Introduction to Functions

Part 2 of 5 — Domain, Range & the Vertical Line Test

🔑 The Idea: The set of all inputs is the domain; the set of all outputs is the range. And a graph is a function exactly when it passes the vertical line test.

Domain and Range

Domain = the set of all inputs ( $x$ -values).
Range = the set of all outputs ( $y$ -values).

Example

For the function ${(1, 4), (2, 5), (3, 6)$ :

Part 3: Function Notation f(x)

⚙️ Introduction to Functions

Part 3 of 5 — Function Notation $f(x)$

🔑 Why $f(x)$ ? The notation $f(x)$ is just a name for the output when the input is $x$ . It is read "" — it does mean times .

Part 4: Reading Tables & Graphs

⚙️ Introduction to Functions

Part 4 of 5 — Reading Tables & Graphs

🔑 Big Idea: A table or graph is a function. You can read an output from an input (go right/up) or read an input from an output (work backward).

Reading a Table

A table lists inputs and their outputs. Here is $f$ :

$x$	$-1$	$0$

Part 5: Mixed Practice & Mastery Check

⚙️ Introduction to Functions

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) tell a function from a relation, (2) find domain and range, (3) use the vertical line test, (4) evaluate $f(x)$ , and (5) read tables and graphs. Let's put it together.

Quick Reference

Idea	What to remember
Function	each input → exactly one output
Domain	the set of inputs ( $x$ -values), each listed once
Range	the set of outputs ( $y$ -values), each listed once

1 \to 3, \quad 2 \to 6, \quad 2 \to 8

1 \to 3, 2 \to 6, 2 \to 8

⚠️ Watch the inputs, not the outputs. Two inputs sharing the same output is perfectly fine. It is only a problem when one input has two outputs.

{(1, 4), (2, 5), (3, 6)}

$\text{Domain} = \{1,\, 2,\, 3\}, \qquad \text{Range} = \{4,\, 5,\, 6\}$

💡 List each value only once. If an output repeats, you still write it a single time in the range. For $\{(1, 7), (2, 7)\}$ the range is just $\{7\}$ .

Find the Range 🧮

Give the range (set of outputs) of each function. Enter the outputs in increasing order, separated by commas, each value once.

1) $\{(0, 2), (1, 5), (2, 8)\}$ → range $= \,?$ 2) $\{(-1, 4), (0, 4), (1, 9)\}$ → range $= \,?$

The Vertical Line Test

On a graph, an input is an $x$ -value and its output is the height $y$ . If a single $x$ had two outputs, the graph would have two points stacked vertically — and a vertical line would cross the graph twice.

🔑 Vertical Line Test (VLT): A graph is a function if every vertical line crosses it at most once.

Graph	Vertical line hits it...	Function?
A straight line $y = 2x + 1$	once	✅ yes
A parabola $y = x^2$	once	✅ yes
A circle $x^2 + y^2 = 9$	twice	❌ no
A sideways parabola $x = y^2$	twice	❌ no

⚠️ A circle fails: the line $x = 0$ hits it at both $(0, 3)$ and $(0, -3)$ — one input, two outputs.

Concept Check 🎯

Bringing It Together

A single function gives you three things at once:

its domain (the inputs you read off the pairs),
its range (the outputs), and
a graph that passes the vertical line test.

The next check asks you to pull all three from one small function.

Domain, Range & VLT 🔽

Use the function $\{(-2, 4), (0, 0), (3, 9)\}$ .

Recap

Domain = inputs, Range = outputs (each value listed once).
Vertical line test: a graph is a function if no vertical line hits it more than once.

Next, in Part 3, we introduce function notation — the famous $f(x)$ — and learn what it really means to "evaluate" a function.

Reading $f(x)$

In $f(x) = 2x + 1$ :

$f$ is the name of the function (the machine).
$x$ is the input (the button you press).
$f(x)$ is the output (what comes out).

To evaluate, substitute the number in for $x$ everywhere it appears.

Example: evaluate $f(3)$ for $f(x) = 2x + 1$

$f(3) = 2(3) + 1 = 6 + 1 = 7$

So $f(3) = 7$ . The point $(3, 7)$ is on the graph of $f$ .

💡 $f(3)$ answers the question: "When the input is $3$ , what is the output?" Here, the answer is $7$ .

Evaluate the Function 🧮

Let $f(x) = 3x - 4$ . Find each output.

1) $f(2) = \,?$ 2) $f(0) = \,?$ 3) $f(-1) = \,?$

A Common Trap

$f(x)$ is not multiplication. If $f(x) = x^2 + 1$ , then:

$f(4) = (4)^2 + 1 = 16 + 1 = 17 \quad\text{(correct)}$

It is not $f \cdot 4$ and it is not $4^2 \cdot 1$ . You substitute $4$ in for $x$ and follow the order of operations.

⚠️ Also be careful with negatives and squaring: $f(-3) = (-3)^2 + 1 = 9 + 1 = 10$ . The square makes the negative positive.

Concept Check 🎯

Outputs Become Points

Every evaluation gives you a point on the graph:

$f(a) = b \quad\Longleftrightarrow\quad \text{the point } (a,\, b) \text{ is on the graph of } f.$

So computing $f(0)$ , $f(1)$ , and $f(-2)$ is the same as finding three points to plot. The next check connects each evaluation to its point.

Match the Notation 🔽

Let $f(x) = 2x + 3$ .

Recap

$f(x)$ = the output for input $x$ ; read " $f$ of $x$ ," never " $f$ times $x$ ."
Evaluate by substituting the input for $x$ and simplifying with order of operations.
$f(a) = b$ means the point $(a, b)$ is on the graph.

In Part 4, we read functions straight from tables and graphs — finding outputs, and even working backward to find inputs.

To find $f(1)$ , look under $x = 1$ : the output is $\mathbf{1}$ , so $f(1) = 1$ .

To work backward — "for which input is $f(x) = 5$ ?" — scan the bottom row for $5$ and read up: it happens at $x = -1$ .

💡 Reading direction matters. $f(x) = ?$ means "go in with $x$ , read out the output." " $f(x) = 5$ , find $x$ " means "start from the output and find the input."

Use the Table 🧮

The same function $f$ :

$x$	$-1$	$0$	$1$	$2$
$f(x)$	$5$	$3$	$1$	$-1$

1) $f(0) = \,?$ 2) $f(2) = \,?$ 3) For which input is $f(x) = 1$ ?

Reading a Graph

On a graph, the point $(a, b)$ means $f(a) = b$ .

Suppose the line $y = f(x)$ passes through $(0, 2)$ , $(1, 4)$ , and $(2, 6)$ .

To find $f(1)$ : go to $x = 1$ , go up to the line, read the height → $f(1) = 4$ .
To solve $f(x) = 6$ : find the height on the line, drop down to the axis → .

These points fit the rule $f(x) = 2x + 2$ . Check: $f(1) = 2(1) + 2 = 4$ ✓ and ✓.

💡 Inputs run horizontally, outputs run vertically. Find an output by moving up to the curve; find an input by moving across to the curve.

Read the Graph 🔽

The line $f(x) = 2x + 2$ passes through $(0, 2)$ , $(1, 4)$ , $(2, 6)$ , and $(3, 8)$ .

Reading Both Directions

Tables and graphs work both ways:

Question	What you do
Find $f(a)$	start at the input $a$ , read off the output
Solve $f(x) = b$	start at the output $b$ , read off the input

Keep straight which value you are given and which you are finding. The next check mixes both directions.

Concept Check 🎯

Recap

A point $(a, b)$ on a graph means $f(a) = b$ .
Find an output: go to the input, move up to the curve, read the height.
Find an input: start from the output's height, move across to the curve, drop to the axis.

In Part 5, we pull everything together with mixed practice and a final Exit Quiz.

⚠️ Repeated outputs are allowed; a repeated input with two different outputs is what breaks a function.

Mixed Practice 🎯

One More Mix

The next check blends all five skills in a single problem: spotting a non-function, evaluating a rule, and applying the vertical line test. Take your time and reason through each piece separately.

Put It Together 🔽

Use $h(x) = x^2 + 1$ and the relation $R = \{(1, 3), (1, 5), (2, 8)\}$ .

You're Ready

You have covered the full arc: relations vs. functions, domain and range, the vertical line test, function notation, and reading tables and graphs. One short quiz stands between you and mastery.

Exit Quiz ✅

Answer all three to finish the lesson.

Introduction to Functions

Introduction to Functions - Complete Interactive Lesson

Part 1: What Is a Function?

⚙️ Introduction to Functions

Topics in This Part

Relations vs. Functions

Spotting a Non-Function

A Quick Way to Test

What You Have So Far

Part 2: Domain, Range & the Vertical Line Test

⚙️ Introduction to Functions

Domain and Range

Example

Part 3: Function Notation f(x)

⚙️ Introduction to Functions

Part 4: Reading Tables & Graphs

⚙️ Introduction to Functions

Reading a Table

Part 5: Mixed Practice & Mastery Check

⚙️ Introduction to Functions

Quick Reference

The Vertical Line Test

Bringing It Together

Recap

Reading f(x)f(x)f(x)

Example: evaluate f(3)f(3)f(3) for f(x)=2x+1f(x) = 2x + 1f(x)=2x+1

A Common Trap

Outputs Become Points

Recap

Reading a Graph

Reading Both Directions

Recap

One More Mix

You're Ready

Reading $f(x)$

Example: evaluate $f(3)$ for $f(x) = 2x + 1$