Relations and Functions

A relation is any rule or set of ordered pairs that connects inputs to outputs. A function is a special relation where every input has exactly one output. Almost everything you’ll study this year is a function, so learning to recognize one is the first step.

Key ideas

Relations

A relation pairs each input $x$ with an output $y$, written as ordered pairs $(x, y)$. You can show the same relation in several ways:

• a list of ordered pairs, like $\{(1, 4), (2, 5), (3, 4)\}$
• a table of values
• a mapping diagram, with arrows from each input to its output(s), like $1 \mapsto 4$
• a graph
• an equation, like $y = x^2$

Functions

A relation is a function if each input has exactly one output.

• Two different inputs can share the same output. That’s fine.
• One input with two different outputs is not allowed.

Think of a vending machine: press B4 and you should get the same snack every time. If B4 sometimes gave chips and sometimes gave a granola bar, the machine wouldn’t be a function.

One-to-one and many-to-one

• One-to-one function: every output comes from just one input. Example: $y = 2x$.
• Many-to-one function: some output comes from more than one input. Example: $y = x^2$, where both $x = 2$ and $x = -2$ give $y = 4$.
• One-to-many (one input, several outputs) is not a function.

The vertical line test

On a graph, an input is an $x$-value, so all the points with that input lie on one vertical line.

A graph is a function if no vertical line crosses it more than once.

The line $x = 1$ crosses $y = x^2 - 2$ once, but crosses $x = y^2 - 2$ twice.

Functions from equations

To check an equation, ask: can one $x$-value give two different $y$-values? A quick way is to try a value of $x$ and solve for $y$.

• $y = x^2 - 2$: each $x$ gives one $y$. Function.
• $x = y^2 - 2$: $x = 2$ gives $y^2 = 4$, so $y = 2$ or $y = -2$. Not a function.

Worked examples

Example 1: Ordered pairs

Is each relation a function?

(a) $\{(1, 4), (2, 5), (3, 4), (4, 7)\}$

(b) $\{(2, 1), (2, 3), (5, 0)\}$

Solution.

(a) The inputs are $1, 2, 3, 4$, and each appears once. It’s a function. The output $4$ appears twice (from $1$ and $3$), which is allowed. It’s a many-to-one function.

(b) The input $2$ is paired with both $1$ and $3$. Not a function.

Example 2: A table of values

$x$	$-1$	$0$	$1$	$2$
$y$	$3$	$1$	$3$	$9$

Is this relation a function? If so, is it one-to-one or many-to-one?

Solution. Each input appears once, so it’s a function. The output $3$ comes from two inputs ($-1$ and $1$), so it is many-to-one.

Example 3: Equations

Which of these are functions?

(a) $y = 3x - 4$ $\qquad$ (b) $x^2 + y^2 = 9$ $\qquad$ (c) $y = \sqrt{x}$

Solution.

(a) Each $x$ gives one value of $3x - 4$. Function (one-to-one; its graph is a slanted line).

(b) Try $x = 0$: $y^2 = 9$, so $y = 3$ or $y = -3$. One input, two outputs. Not a function. (Its graph is a circle, which fails the vertical line test.)

(c) $\sqrt{x}$ means the positive square root, so each $x \ge 0$ gives exactly one $y$. Function.

Common mistakes

Thinking a repeated output breaks the rule. $\{(1, 5), (2, 5)\}$ is a function. The rule is about inputs: one input can’t have two outputs.

Using a horizontal line instead of a vertical one. The vertical line test checks for a function. A horizontal line test checks something else (whether the function is one-to-one), which matters later for inverse functions.

Worrying about a repeated pair. If $(2, 5)$ is listed twice, it’s still the same input with the same output. That’s fine.

Assuming every equation is a function. Circles, like $x^2 + y^2 = 9$, and sideways parabolas, like $x = y^2$, are relations but not functions. Test a value of $x$ when you’re unsure.

Practice

1. (Warm-up) Is $\{(-3, 2), (0, 2), (4, 2)\}$ a function?

Solution

Yes. Each input ($-3$, $0$, $4$) has exactly one output. All three share the output $2$, which is allowed (many-to-one).

2. (Warm-up) Is $\{(1, 5), (3, -2), (1, 7), (6, 0)\}$ a function?

Solution

No. The input $1$ is paired with both $5$ and $7$.

3. (Warm-up) The points $(2, 5)$ and $(2, -1)$ are both on a graph. Can the graph be a function?

Solution

No. Both points have the input $x = 2$ but different outputs, so the vertical line $x = 2$ crosses the graph twice.

4. (Core) Is this relation a function? If so, is it one-to-one or many-to-one?

$x$	$-2$	$-1$	$0$	$1$	$2$
$y$	$4$	$1$	$0$	$1$	$4$

Solution

Each input appears once, so it’s a function. Outputs $4$ and $1$ each come from two inputs, so it’s many-to-one. (These values fit $y = x^2$.)

5. (Core) Decide whether each equation is a function.

• (a) $y = 2x^2 - 1$
• (b) $y^2 = x + 4$
• (c) $y = \dfrac{1}{x}$
• (d) $x = 3$

Solution

(a) Function. Each $x$ gives one value of $2x^2 - 1$.

(b) Not a function. For $x = 0$, $y^2 = 4$, so $y = 2$ or $y = -2$.

(c) Function. Each $x \ne 0$ gives one value of $\dfrac{1}{x}$. It’s one-to-one.

(d) Not a function. Every point on this graph has $x = 3$, so the single input $3$ has infinitely many outputs. (The graph is itself a vertical line.)

6. (Core) Is each relation a function? Explain.

• (a) Each student in your class is paired with the month they were born in.
• (b) Each month is paired with the students in your class born in that month.

Solution

(a) Function. Every student has exactly one birth month. Many students can share a month, so it’s many-to-one.

(b) Not a function (in most classes). A month like March could be paired with several students, so one input has several outputs.

7. (Challenge) For which values of $k$ is $\{(k, 2), (3, 5), (1, 2)\}$ a function?

Solution

The only problem would be the input $3$ having two different outputs. That happens if $k = 3$, because then $(3, 2)$ and $(3, 5)$ are both in the relation.

If $k = 1$, the pair $(1, 2)$ just appears twice, which is fine.

So the relation is a function for every value of $k$ except $3$.

8. (Challenge) The circle $x^2 + y^2 = 25$ is not a function, but its top half, $y = \sqrt{25 - x^2}$, is. Explain why.

Solution

Solving the circle’s equation for $y$ gives $y = \pm\sqrt{25 - x^2}$. For any $x$ between $-5$ and $5$, that’s two outputs (one positive, one negative), so the circle fails the vertical line test.

The top half keeps only the positive square root, $y = \sqrt{25 - x^2}$. Now each $x$ from $-5$ to $5$ gives exactly one $y$, so every vertical line crosses it at most once.