Introduction to Functions

⚙️ Introduction to Functions

Part 1 of 5 — What Is a Function?

Topics in This Part

🔑 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.

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:

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

⚠️ 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.

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:

1. List every input (first coordinate).
2. If the same input ever shows up with two different outputs, it is not a function.
3. 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.

⚙️ 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 $\{(1, 4), (2, 5), (3, 6)\}$:

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

💡 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.

⚠️ 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.

⚙️ 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 "f of x" — it does not mean $f$ times $x$.

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.

⚙️ 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 $f$:

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 $f$:

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

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 $6$ on the line, drop down to the axis → $x = 2$.

These points fit the rule $f(x) = 2x + 2$. Check: $f(1) = 2(1) + 2 = 4$ ✓ and $f(2) = 2(2)+2 = 6$ ✓.

💡 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:

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.

⚙️ 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

⚠️ 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.