Topic 2 of 7

Functions, domain, and range

Step 11 · Algebra 1 topics

Decide whether a relation is a function, use function notation f(x), and find the domain and range of a function from its formula, its graph or its context.

What makes a function

A function gives exactly one output for each input. It can be given as a rule, a table, a list of pairs or a graph.

On a graph, the vertical line test checks this: if any vertical line crosses the graph twice, one input has two outputs, and it is not a function.

y = x² passes the vertical line test: the dashed line, like every vertical line, meets it only once.

Example: Testing a relation

Is x = y² a function of x?

When x = 4, y can be 2 or −2.
One input has two outputs.

AnswerNo

Function notation

f(x), read "f of x", is the output of the function f when the input is x. It does not mean f times x.

Example: Evaluating

f(x) = 3x² − 2x. Find f(−2).

Replace every x with (−2): 3(−2)² − 2(−2).
3 × 4 + 4 = 16.

Answer16

Example: Solving f(x) = k

g(x) = 5x − 3. For which x is g(x) = 17?

5x − 3 = 17.
5x = 20, so x = 4.

Answerx = 4

Example: An expression as the input

f(x) = x + 4. Find f(a + 1).

Put a + 1 in place of x: (a + 1) + 4.

Answera + 5

Domain and range

The domain is the set of inputs a function can take. The range is the set of outputs it actually gives.

Function	Domain	Range
f(x) = 2x + 1	all real numbers	all real numbers
f(x) = x²	all real numbers	y ≥ 0
f(x) = √x	x ≥ 0	y ≥ 0
f(x) = $\frac{1}{x}$	x ≠ 0	y ≠ 0

Example: A square root

What is the domain of f(x) = √(x − 3)?

The number under the root cannot be negative: x − 3 ≥ 0.

Answerx ≥ 3

Example: From a graph

A graph is a solid segment from (−2, 1) to (4, 7). What are its domain and range?

The inputs run from −2 to 4.
The outputs run from 1 to 7.

AnswerDomain −2 ≤ x ≤ 4, range 1 ≤ y ≤ 7

Try it yourself

Work each one on paper first, then open the answer to check.

Is {(1, 3), (2, 3), (3, 3)} a function?
Show answer
Yes: each input has one output.
f(x) = x² − 4x. Find f(5).
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5
f(x) = 2x + 7. For which x is f(x) = 1?
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x = −3
f(x) = x − 2. Find f(3a).
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3a − 2
What is the domain of f(x) = √(x + 5)?
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x ≥ −5
What is the range of f(x) = x² + 1?
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y ≥ 1
What is the domain of $g(x) = \frac{3}{x - 2}$ ?
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All real numbers except 2