Functions: f(x) notation, domain and range explained

The key idea

Think of a function as a machine. You drop in an input, it follows its rule, and one output comes out. The name of the machine is f, and $f(x)$ is read "f of x." It does not mean f times x.

f(x) = 2x^2 - 3 \quad\Longrightarrow\quad f(-2) = 2(-2)^2 - 3 = 5

The words you need
Word	Meaning	Example for $f(x) = \sqrt{x}$
input	the x value you put in	9
output	the value that comes out, $f(x)$ or y	$f(9) = 3$
domain	all allowed inputs	$x \ge 0$ , because you cannot take the square root of a negative number
range	all possible outputs	$y \ge 0$ , because a square root is never negative

In Algebra 1, two things shrink a domain: you cannot divide by zero, and you cannot take the square root of a negative number. If neither shows up, the domain is all real numbers. In a word problem, the situation can limit it too, like not having a negative number of tickets.

Worked examples

Example 1: evaluate a function

Problem If $f(x) = x^2 - 4x + 1$ , find $f(-3)$ .

Replace every x with $(-3)$ , in parentheses. Then follow the order of operations.
$(-3)^2 - 4(-3) + 1 = 9 + 12 + 1 = 22$
The parentheses matter: $(-3)^2 = 9$ , while $-3^2$ would mean $-(3^2) = -9$ .

Answer $f(-3) = 22$

Example 2: find the input that gives an output

Problem If $g(x) = 3x - 7$ , for what value of x is $g(x) = 11$ ?

This time the output is given. Set the rule equal to 11.
$3x - 7 = 11$
Add 7.
$3x = 18$
Divide by 3.
$x = 6$
Check: $g(6) = 18 - 7 = 11$ .

Answer $x = 6$

Example 3: domain and range from an equation

Problem Find the domain and range of $h(x) = \sqrt{x - 5}$ .

The expression under the square root cannot be negative.
$x - 5 \ge 0$
Add 5. That is the domain.
$x \ge 5$
The smallest output is $h(5) = \sqrt{0} = 0$ . As x grows, the square root grows without limit, and it is never negative. So the range is $y \ge 0$ .

Answer Domain: $x \ge 5$ . Range: $y \ge 0$ .

Example 4 (test-hard): an expression as the input

Problem If $f(x) = x^2 + 2x$ , find and simplify $f(t - 1)$ .

Replace every x with $(t - 1)$ , then expand and combine like terms.
$(t - 1)^2 + 2(t - 1) = t^2 - 2t + 1 + 2t - 2 = t^2 - 1$
Check with a number. If $t = 3$ , then $f(2) = 4 + 4 = 8$ , and $3^2 - 1 = 8$ . They agree.

Answer $f(t - 1) = t^2 - 1$

Common mistakes

Reading f(x) as multiplication. $f(3)$ is the output when the input is 3. It is not $f \cdot 3$ .
Leaving out parentheses for a negative input. $f(-3)$ for $x^2$ is $(-3)^2 = 9$ . Typing $-3^2$ into a calculator gives $-9$ .
Mixing up f(3) and f(x) = 3. The first gives you the input and asks for the output. The second gives you the output and asks for the input.
Calling the x values the range. Domain is inputs (x). Range is outputs (y).
Thinking a repeated output breaks the rule. $\{(1, 2), (2, 2)\}$ is a function. Only a repeated input with two different outputs, like $(0, 5)$ and $(0, 7)$ , is not.

Quick methods

Practice

5 practice questions

If $f(x) = 5 - 2x$ , what is $f(-4)$ ?
1. $-3$
2. $13$
3. $-13$
4. $22$
Show answer
Answer: $13$
$f(-4) = 5 - 2(-4) = 5 + 8 = 13$ . The answer $-3$ comes from using 4 instead of $-4$ .
Which set of ordered pairs is NOT a function?
1. $\{(1, 2), (2, 2), (3, 2)\}$
2. $\{(0, 5), (1, 6), (0, 7)\}$
3. $\{(-1, 1), (1, 1), (2, 4)\}$
4. $\{(4, 0), (5, 1), (6, 2)\}$
Show answer
Answer: $\{(0, 5), (1, 6), (0, 7)\}$
The input 0 gives two different outputs, 5 and 7, so it breaks the one-output rule. The other sets repeat some outputs, which is allowed.
What is the domain of $g(x) = \frac{1}{x - 3}$ ?
1. all real numbers
2. all real numbers except 3
3. all real numbers except $-3$
4. $x > 3$
Show answer
Answer: all real numbers except 3
The denominator is 0 when $x = 3$ , and you cannot divide by zero. Every other number works, including numbers less than 3, so $x > 3$ is too strict.
If $f(x) = x^2 - 3x$ , which value of x makes $f(x) = 10$ ?
1. $2$
2. $5$
3. $10$
4. $70$
Show answer
Answer: $5$
Test it: $f(5) = 25 - 15 = 10$ . The trap answer 70 is $f(10)$ : it treats 10 as the input instead of the output. (The equation also has a second solution, $x = -2$ , which is not listed.)
What is the range of $h(x) = x^2 + 1$ ?
1. all real numbers
2. $y \ge 0$
3. $y \ge 1$
4. $y > 1$
Show answer
Answer: $y \ge 1$
$x^2$ is never negative, so the smallest output is $0 + 1 = 1$ , reached at $x = 0$ . Every larger value is reached too. $y > 1$ wrongly leaves out 1 itself.

Frequently asked questions

What does f(x) mean?

It means "the output of the function f when the input is x." The letter f names the function, and whatever is in the parentheses is the input. So $f(4)$ is the output when you put in 4. It is a label, not multiplication.

How do I find the domain of a function?

Start with all real numbers, then remove anything that breaks a rule. Remove inputs that make a denominator zero. Keep only inputs that make the expression under a square root zero or positive. In a word problem, also remove values that make no sense, like negative times.

What is the difference between domain and range?

The domain is the set of inputs, the x values. The range is the set of outputs, the y values. A memory trick: d comes before r in the alphabet, just as x comes before y in the pair $(x, y)$ .

How do I know if a graph is a function?

Use the vertical line test. If any vertical line touches the graph more than once, some input has two outputs, so it is not a function. If every vertical line touches it at most once, it is a function.

Functions, function notation, domain and range

The key idea

Worked examples

Common mistakes

Quick methods

Practice

Frequently asked questions

Try asking Ducky

The key idea

Worked examples

Common mistakes

Quick methods

Practice

Frequently asked questions

Related

Try asking Ducky