Before you read

Answer first — see what you already know.

What is the domain of $f (x) = 5 x + 3$ ?

A
$x \geq 0$
B
All real numbers
C
$x \geq 3$
D
$x \neq = 0$

Check your answer

Answer:

Before you read

Answer first — see what you already know.

A swimming pool is being filled. The depth $d$ (in feet) after $t$ hours is $d = 0.5 t + 1$ . The pool's maximum depth is $6$ feet. What is the domain of this function in context?

A
$0 \leq t \leq 6$
B
$0 \leq t \leq 10$
C
$t \geq 0$
D
All real numbers

Check your answer

Answer:

Theory

Domain and Range — Definitions

Domain = all possible input values ( $x$ -values)
Range = all possible output values ( $y$ -values)

Think of it this way:
- Domain: "What can I put in?"
- Range: "What can come out?"

For a pure linear function $f(x) = mx + b$ with no restrictions:
- Domain = all real numbers ( $-\infty < x < \infty$ )
- Range = all real numbers ( $-\infty < y < \infty$ )

f(x) = x: domain and range are all real numbers

But in real-world contexts, there are usually restrictions!

Theory

Domain and Range from Context

Real-world problems often restrict the domain:

Context	Domain restriction	Why
Time	$t \geq 0$	Time can't be negative
Quantity/count	$n \geq 0$ , integers	Can't have $-3$ items
Distance	$d \geq 0$	Distance isn't negative
Percentage	$0 \leq p \leq 100$	Bounded

The range then depends on the domain restriction combined with the function rule.

Example 1

A taxi ride costs $C = 3.50 + 2m$ dollars, where $m$ is miles. What are the domain and range?

Domain: $m \geq 0$ (you can't drive negative miles)

Range: When $m = 0$ , $C = 3.50$ . As $m$ increases, $C$ increases.

Range: $C \geq 3.50$

Domain:

m \geq 0

, Range:

C \geq 3.50

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