Functions and function notation

Step 1 of 5Warm-up

Two quick questions from earlier lessons. Answer from memory first. Pulling it out of your head is what makes it stick.

Solve $x^2 - x - 6 \le 0$. Give the answer in interval notation.

Factor: $(x - 3)(x + 2) \le 0$. The zeros are $x = 3$ and $x = -2$.

The parabola opens up, so it is below zero between the zeros. The zeros themselves count because of the $\le$.

Answer: $[-2, 3]$.

Evaluate $2x^2 - 3x + 1$ when $x = -2$.

Put $(-2)$ in for every $x$: $2(-2)^2 - 3(-2) + 1$.

$(-2)^2 = 4$, so this is $8 + 6 + 1 = 15$.

Step 2 of 5The idea

A relation is any set of input-output pairs, written as ordered pairs (input, output). A function is a relation with one extra promise: each input has exactly one output.

So $\{(1, 4), (2, 5), (1, 6)\}$ is not a function: the input 1 has two outputs, 4 and 6. But $\{(1, 4), (2, 4)\}$ is a function. Two inputs sharing an output is fine.

On a graph, use the vertical line test. If some vertical line hits the graph twice, one input has two outputs, so it is not a function.

The line $x = 4$ hits this graph twice, so it is not a function.

Function notation $f(x)$ is read "f of x". It means "the output of $f$ when the input is $x$". It is not $f$ times $x$.

Function: every input has exactly one output.

Vertical line test: a graph is a function if no vertical line crosses it more than once.

To find $f(a)$: replace every $x$ in the formula with $(a)$, in parentheses, then simplify.

From a table or graph: $f(2)$ is the output (the y-value) that goes with the input $x = 2$.

Why this shows up on the placement test: function notation is the language of College Algebra, and "find $f(-3)$" is one of the most common quick questions.

Step 3 of 5Worked examples

Example 1: $f(x) = x^2 - 3x + 2$. Find $f(-2)$ and $f(a + 1)$.

Replace each $x$ with $(-2)$: $f(-2) = (-2)^2 - 3(-2) + 2$.
$= 4 + 6 + 2 = 12$. $(-2)^2 = 4$ and $-3 \cdot (-2) = +6$.
Now replace each $x$ with $(a + 1)$: $f(a + 1) = (a + 1)^2 - 3(a + 1) + 2$.
Expand: $a^2 + 2a + 1 - 3a - 3 + 2$. $(a+1)^2 = a^2 + 2a + 1$, and distribute the $-3$.
Combine like terms: $f(a + 1) = a^2 - a$.

Example 2: reading a function from its graph

The graph of a function $f$.

To find $f(0)$, start at $x = 0$ on the x-axis and go up to the graph. You land at height 3, so $f(0) = 3$.
To find $f(2)$, go to $x = 2$. The graph is at height $-1$, so $f(2) = -1$.
Now go the other way: which inputs give $f(x) = 3$? Look across at height 3. The graph is there at $x = 0$ and $x = 4$. Two inputs can share one output. It still passes the vertical line test.

Step 4 of 5Practice

Type your answer and press Check. If you miss twice, open the worked solution. Then mark how it went.

$f(x) = 3x - 5$. Find $f(4)$.

$f(4) = 3(4) - 5 = 12 - 5 = 7$.

$g(x) = x^2 + 2x - 1$. Find $g(-3)$.

$g(-3) = (-3)^2 + 2(-3) - 1$.

$= 9 - 6 - 1 = 2$.

Which relation is a function?

$\{(2, 1), (3, 4), (2, 5)\}$
$\{(0, 3), (1, 3), (0, -3)\}$
$\{(4, 1), (4, 2), (5, 3)\}$
$\{(1, 6), (2, 6), (3, 6)\}$

Look only at the inputs (the first numbers). A repeated input with different outputs breaks the rule.

(a) Input 2 has outputs 1 and 5. Not a function.

(b) Input 0 has outputs 3 and $-3$. Not a function.

(c) Input 4 has outputs 1 and 2. Not a function.

(d) Inputs 1, 2, 3 each appear once. They share the output 6, which is allowed. This one.

Use the table. For which input $x$ is $f(x) = -3$?

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

Here you know the output, $-3$, and want the input. Find $-3$ in the $f(x)$ row.

It sits under $x = 2$. So $f(2) = -3$, and the answer is $x = 2$.

Use the graph of $f$ below. Find $f(-2)$.

The graph of $f$. Each grid square is 1 unit.

Go to $x = -2$ on the x-axis, then straight up to the graph.

The graph is at height 1 there. So $f(-2) = 1$.

$f(x) = 2x^2 - x$. Find $f(a + 1)$ and simplify.

Replace each $x$ with $(a + 1)$: $2(a + 1)^2 - (a + 1)$.

$(a + 1)^2 = a^2 + 2a + 1$, so this is $2a^2 + 4a + 2 - a - 1$.

Combine like terms: $2a^2 + 3a + 1$.

$f(x) = 4x - 7$. Find the input $x$ that makes $f(x) = 9$.

Set the formula equal to 9: $4x - 7 = 9$.

Add 7: $4x = 16$. Divide by 4: $x = 4$.

Check: $f(4) = 16 - 7 = 9$.

A plumber's bill is $C(t) = 45 + 30t$ dollars for a job that takes $t$ hours. Find $C(2.5)$, the bill for a job of 2.5 hours.

$C(2.5) = 45 + 30(2.5)$.

$30 \cdot 2.5 = 75$, so $C(2.5) = 45 + 75 = 120$. The bill is $120.

Step 5 of 5Wrap-up

You can now spot a function (one output per input, or the vertical line test) and read $f(x)$ as "the output for input $x$", from a formula, a table, or a graph.

Read or watch next: OpenStax Intermediate Algebra 2e, 3.5 Relations and Functions (free textbook, with more practice at the end of the section).

Also good: Khan Academy: Algebra 1: Evaluating functions (short videos). Quick rules: Functions cheat sheet.

Something unclear? Ask your teacher bot a follow-up question, for example: "Why is it OK for two inputs to have the same output, but not one input with two outputs?" It knows this course and can explain things another way.

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