What Is a Function?

Lesson 1 of 3: Functions

In this lesson:

• What makes a rule a function
• Functions as ordered pairs and graphs

Learning Objectives: Defining a Function

By the end of this lesson, you should be able to:

1. Define a function as a rule giving each input exactly one output
2. Decide whether a table, set of pairs, or graph is a function
3. Represent a function as ordered pairs and plot them
4. Apply the vertical line test to a graph
5. Explain why two outputs for one input breaks the rule

Same Button, Same Snack Every Time?

You press B4 on a vending machine and a granola bar drops out.

Press B4 again tomorrow.

Do you expect the same granola bar — or a surprise?

Feed Numbers Through the Machine

Our rule: multiply the input by 2, then add 1.

Put in 3 again. What comes out?

This Reliable Rule Has a Name

A rule like this — where every input you feed in produces a single, predictable output — is called a function.

The machine is trustworthy: 3 always gives 7.

The Definition of a Function

A function is a rule that assigns to each input exactly one output.

• The word that matters most is exactly one
• Inputs, outputs, and rules — no special symbols needed yet

When the Machine Breaks Its Promise

A "mystery box" gives these results:

Is this a function? Look hard at input 2.

Sharing an Output Is Fine

Every student has exactly one birthday month.

Two students can share March — that's allowed!

So is a function.

Quick Check: Function or Not?

Table A: 1→4, 2→8, 3→4

Table B: 5→1, 6→2, 5→9

Which one is a function? Decide before advancing.

From Table to Ordered Pairs

Rewrite each row of our rule as a pair :

The input is the -coordinate; the output is the -coordinate.

Plotting a Function as Points

Each point sits at its own — the points line up.

What a Non-Function Looks Like

• Both points sit at
• Input 2 has two outputs — stacked vertically

Quick Check: Which Is a Function?

Set P:

Set Q:

Which set is a function? Look for a repeated .

Stacked Points Reveal the Test

When a relationship is not a function, two of its points stack at the same .

So a vertical line through that hits the graph twice.

That stack is about to become a test for any graph.

The Graph and the Vertical Line Test

The graph of a function is the set of all its ordered pairs.

Vertical line test: if any vertical line crosses the graph more than once, it is not a function.

Sweeping a Line and a Parabola

• The line: one hit at every — function
• The parabola: one hit at every — function

Sweeping a Vertical Line on a Circle

• At , the line hits top and bottom
• Input 1 has two outputs — not a function

Is a Vertical Line a Function?

The graph is a straight, vertical line.

It's a line — so it must be a function, right? Predict before advancing.

Why Is Not a Function

Lay the test line on — it overlaps the whole graph.

Input 4 produces every output at once.

Compare: (horizontal) is a function — every input gives 4.

Find the Error in This Claim

A student claims:

" is a function — all the outputs are different."

What did they get wrong? Name the input that breaks it.

Your Turn: Decide on Your Own

On your whiteboard, classify this graph:

A single straight slanted line crossing the plane.

Write "function" or "not" — and one reason citing the definition.

Classify All Three Relationships Now

Decide function or not — justify each:

1. Table:
2. Pairs:
3. Graph: a sideways U opening right

Three Common Traps to Avoid

Same output is fine — only repeated inputs with different outputs break it

is never a function; always is

A table or word-rule is a function — no formula required

One Promise Seen in Three Views

✓ A function keeps one promise: each input, exactly one output

✓ See it in a table, in points, and on a graph

Next: comparing two functions — which one is bigger?