Domain and Range
Domain and Range
The domain of a function is the set of all allowable inputs, and the range is the set of all possible outputs. Identifying domain and range is essential because it tells you where the function "lives" and what values it can produce.
For most functions studied in algebra, the domain is restricted only by two dangers: dividing by zero and taking the square root of a negative number. Recognizing and excluding these trouble spots is the central skill of this lesson.
You will learn to state domain and range using inequality notation, interval notation, and set-builder notation, and to read them from graphs, equations, and tables.
Domain: all possible input ($x$) values.
Range: all resulting output ($y$) values.
Exclude $x$-values that cause:
- Division by zero: set denominator $\neq 0$
- Even root of negative: set radicand $\ge 0$
Interval Notation
| Notation | Meaning |
|---|---|
| $(a, b)$ | $a < x < b$ (open) |
| $[a, b]$ | $a \le x \le b$ (closed) |
| $[a, b)$ | $a \le x < b$ |
| $(-\infty, b]$ | $x \le b$ |
| $(-\infty, \infty)$ | all real numbers |
Domain of $f(x) = \dfrac{1}{x - 3}$.
$x - 3 \neq 0 \Rightarrow x \neq 3$. Domain: $(-\infty, 3) \cup (3, \infty)$.
Domain of $g(x) = \sqrt{x + 2}$.
$x + 2 \ge 0 \Rightarrow x \ge -2$. Domain: $[-2, \infty)$.
Domain of $h(x) = \dfrac{\sqrt{x}}{x - 4}$.
Need $x \ge 0$ (square root) and $x \neq 4$ (denominator). Domain: $[0, 4) \cup (4, \infty)$.
Range of $f(x) = x^2$.
$x^2 \ge 0$ always. Range: $[0, \infty)$.
Domain and range of $f(x) = |x - 1| + 2$.
Domain: $(-\infty, \infty)$. Since $|x - 1| \ge 0$, we have $f(x) \ge 2$. Range: $[2, \infty)$.
Practice Problems
Find the domain of each function.
Show Answer Key
1. $(-\infty, \infty)$
2. $x \neq -1$; $(-\infty, -1) \cup (-1, \infty)$
3. $[5, \infty)$
4. $x \neq \pm 3$
5. $(-\infty, 4]$
6. $(-\infty, \infty)$
7. Domain: $(-\infty, \infty)$; Range: $(0, 1]$
8. $(-\infty, 4]$
9. $[-3, 2) \cup (2, \infty)$
10. $[1, \infty)$