Domain and Range

Two terms come up constantly when you work with functions: domain and range. They describe the boundaries of what a function can take in and what it can put out.

What Is the Domain?

The domain of a function is the set of all valid inputs — every value you're allowed to plug in. For most simple algebraic functions, the domain is all real numbers, because there's nothing that would break the function.

Take $f(x) = 2x + 1$. What can you plug in for $x$? Anything. Negative numbers, zero, fractions, billion-digit decimals — they all produce a valid output. The domain is all real numbers.

Graph of a line with unrestricted domain

Visually, that's a line that stretches forever in both horizontal directions. There's no $x$-value that the line fails to cover.

Common Domain Restrictions

A few situations break a function and force you to exclude certain inputs from the domain.

Division by Zero

The most common culprit. Consider:

$$y = \frac{3}{x - 1}$$

For almost every $x$, this works fine. But what happens at $x = 1$? The denominator becomes zero, and $\frac{3}{0}$ is undefined. So $x = 1$ is excluded from the domain. The domain is all real numbers except $x = 1$.

When looking for domain restrictions, the first thing to check is whether any $x$-value makes a denominator equal to zero. Those values get tossed out.

Square Roots of Negative Numbers

In the real numbers, you can't take the square root of a negative. So if a variable appears inside a square root (or any even-indexed radical), the expression underneath must be zero or positive.

Example: Find the domain of $f(x) = \sqrt{x - 2}$.

The expression inside the radical, $x - 2$, must be $\geq 0$. That gives $x \geq 2$. The domain is all real numbers greater than or equal to 2.

Other Restrictions

Inverse trig functions have built-in restrictions: since the output of $\sin x$ is always between $-1$ and $1$, the input of $\arcsin x$ is also restricted to that interval. Logarithms only accept positive inputs. These come up later in algebra and precalculus, but the same general idea applies — identify any input that "breaks" the function, and exclude it.

When asked to find the domain, the standard checklist is: any denominators that could equal zero, any radicals that could go negative, and any other operation that has a built-in input restriction.

What Is the Range?

The range is the set of all possible outputs — every value of $y$ (or $f(x)$) the function can produce.

For most simple linear functions, the range is also all real numbers. A line like $y = 3x$ extends forever in both vertical directions:

Graph of y = 3x with range all real numbers

There's no $y$-value the line fails to hit, so the range is all real numbers.

The clear exception is a horizontal line. The function $y = 3$ produces only one output (the number 3), no matter what $x$ is:

Graph of horizontal line y = 3 with range just {3}

The range of $y = 3$ is just ${3}$. One output and only one output.

Functions with Limited Ranges

Functions with restricted ranges are much more common than functions with restricted domains. Two examples:

Graphs of y = x² - 2 (blue) and y = sin x (red) showing limited ranges

The blue parabola is $y = x^2 - 2$. Because $x^2$ is always zero or positive, the smallest value of $y$ is $-2$ (reached at $x = 0$). The function never produces an output below $-2$, so the range is $y \geq -2$.

The red curve is $y = \sin x$. Sine oscillates between $-1$ and $1$ forever, never reaching beyond. The range is $-1 \leq y \leq 1$.

When checking the range of a function, two approaches work. Looking at the graph and reading off the lowest and highest $y$-values is the fastest visual check. Algebraically, watch for operations that constrain the output: even powers ($x^2$, $x^4$) always produce non-negative results, absolute values are always non-negative, and trigonometric functions have built-in bounds.

Summary

The domain is all the possible inputs the function will accept; the range is all the possible outputs it produces.

Diagram illustrating domain and range as input and output sets

Most algebraic functions have a domain of all real numbers unless a denominator could be zero or a radical could go negative. Most functions have a more restricted range, especially anything involving squares, absolute values, or trigonometric operations.

To check the domain and range of any function you enter, try the Domain and Range Calculator — it walks through the restrictions step by step and reports both sets in interval notation.

Practice Problems

1. Find the domain of $f(x) = \dfrac{5}{x + 3}$. Show answerThe denominator is zero when $x = -3$, so that value is excluded. Domain: all real numbers except $x = -3$.

2. Find the domain of $g(x) = \sqrt{x + 4}$. Show answerThe expression under the radical must be non-negative: $x + 4 \geq 0$, so $x \geq -4$. Domain: $x \geq -4$.

3. Find the domain of $h(x) = \dfrac{x}{x^2 - 9}$. Show answerFactor the denominator: $x^2 - 9 = (x-3)(x+3)$. This is zero at $x = 3$ and $x = -3$, so both are excluded. Domain: all real numbers except $x = \pm 3$.

4. What is the range of $y = x^2 + 1$? Show answer$x^2$ is always $\geq 0$, so $x^2 + 1$ is always $\geq 1$. Range: $y \geq 1$.

5. What is the range of $y = -|x|$? Show answer$|x|$ is always $\geq 0$, so $-|x|$ is always $\leq 0$. Range: $y \leq 0$.

6. What is the domain and range of $y = 7$? Show answerAny $x$ is valid, so the domain is all real numbers. The output is always 7, so the range is ${7}$.