Domain and Range Calculator

Free Online Domain and Range Calculator

Our Domain and Range Calculator helps you find the domain and range of any function with step-by-step analysis. Whether you are studying algebra, calculus, or preparing for exams, this tool provides instant results with interval notation, KaTeX math rendering, and detailed explanations.

What is the Domain of a Function?

The domain of a function $f(x)$ is the set of all possible input values ($x$) for which the function produces a valid output. It represents all the $x$-values you can substitute into the function without causing mathematical errors such as division by zero or taking the square root of a negative number.

Common restrictions that limit the domain include:

Division by zero: The denominator of a fraction cannot equal zero
Even roots of negative numbers: Square roots, fourth roots, etc. require non-negative radicands in real numbers
Logarithms: The argument of a logarithm must be positive
Inverse trigonometric functions: $\arcsin(x)$ and $\arccos(x)$ have specific input restrictions ($-1 \le x \le 1$)

What is the Range of a Function?

The range of a function $f(x)$ is the set of all possible output values ($y$) that the function can produce. It represents all the values $f(x)$ can actually attain as $x$ varies over the domain. Finding the range often requires analyzing maximum and minimum values, asymptotic behavior, and function transformations.

Common Function Types and Their Domain/Range

Function Type	Example	Domain	Range
Linear	$f(x) = mx + b$	$(-\infty, \infty)$	$(-\infty, \infty)$
Quadratic	$f(x) = ax^2 + bx + c$	$(-\infty, \infty)$	$[k, \infty)$ or $(-\infty, k]$
Square Root	$f(x) = \sqrt{x}$	$[0, \infty)$	$[0, \infty)$
Rational	$f(x) = \frac{1}{x}$	$(-\infty, 0) \cup (0, \infty)$	$(-\infty, 0) \cup (0, \infty)$
Logarithmic	$f(x) = \log(x)$	$(0, \infty)$	$(-\infty, \infty)$
Exponential	$f(x) = e^x$	$(-\infty, \infty)$	$(0, \infty)$
Sine	$f(x) = \sin(x)$	$(-\infty, \infty)$	$[-1, 1]$
Absolute Value	$f(x) = \|x\|$	$(-\infty, \infty)$	$[0, \infty)$

How to Find Domain — Step by Step

Step 1: Identify Potential Restrictions

Look for operations that have input restrictions: fractions (denominators cannot equal zero), even roots (radicand must be non-negative), and logarithms (argument must be positive).

Step 2: Solve for Restricted Values

For each restriction identified, solve the equation or inequality to find the excluded values.

Step 3: Write the Domain in Interval Notation

Express the domain using interval notation, excluding the restricted values. Use parentheses $(\;)$ for open intervals (value not included) and brackets $[\;]$ for closed intervals (value included).

How to Use the Domain and Range Calculator

Enter your function in the input field using standard mathematical notation.
Use standard operations like +, -, *, /, ^ for exponentiation.
Available functions: sqrt(x), log(x), ln(x), sin(x), cos(x), tan(x), abs(x), exp(x).
Click an example or type your own function to see instant results.
Review the domain and range displayed with KaTeX math notation and a step-by-step breakdown.

Common Scenarios for Domain Restrictions

Division by zero: For rational functions, exclude $x$-values that make the denominator zero. Example: $f(x) = \frac{1}{x-3}$ has domain $x \neq 3$.
Even roots of negatives: For square roots and other even roots, the radicand must be $\ge 0$. Example: $f(x) = \sqrt{x-2}$ has domain $x \ge 2$.
Logarithms of non-positive numbers: For $\ln(x)$ and $\log(x)$, the argument must be $> 0$. Example: $f(x) = \ln(x+5)$ has domain $x > -5$.
Tangent function: $f(x) = \tan(x)$ excludes $x = \frac{\pi}{2} + n\pi$ where $n$ is an integer, because $\cos(x) = 0$ at those points.

Interval Notation Guide

$(a, b)$: Open interval — all numbers between $a$ and $b$, excluding endpoints.
$[a, b]$: Closed interval — all numbers between $a$ and $b$, including endpoints.
$[a, b)$: Includes $a$ but excludes $b$.
$(-\infty, a]$: All numbers less than or equal to $a$.
$(a, \infty)$: All numbers greater than $a$.
$\cup$: Union symbol — used to combine multiple intervals.
$\emptyset$: Empty set — no solution exists.

Frequently Asked Questions

How is domain different from range?

Domain refers to all possible input values ($x$-values), while range refers to all possible output values ($y$-values). Think of domain as "what you can put in" and range as "what you can get out." For example, for $f(x) = \sqrt{x}$, the domain is $[0, \infty)$ and the range is also $[0, \infty)$.

Can a function have an empty domain?

Yes, a function can have an empty domain if there are no real values of $x$ that make the function defined. For example, $f(x) = \sqrt{-x^2-1}$ has no real domain because $-x^2-1$ is always negative, and you cannot take the square root of a negative number in the real number system.

Why is infinity always written with parentheses?

Infinity is always written with parentheses because it is not a real number that can be reached or included. We can only approach infinity, never actually include it in an interval. For example, $[3, \infty)$ means all numbers greater than or equal to 3, but infinity itself is never included.

How do I find the domain of a rational function?

For rational functions (fractions with polynomials), set the denominator equal to zero and solve for $x$. The domain is all real numbers except those values. For example, for $f(x) = \frac{1}{x^2-4}$, set $x^2-4 = 0$, giving $x = 2$ and $x = -2$. The domain is $(-\infty, -2) \cup (-2, 2) \cup (2, \infty)$.

What is the domain and range of a quadratic function?

The domain of any quadratic function $f(x) = ax^2 + bx + c$ is all real numbers $(-\infty, \infty)$. The range depends on the sign of $a$: if $a > 0$, the parabola opens upward and the range is $[k, \infty)$ where $k$ is the $y$-coordinate of the vertex; if $a < 0$, the range is $(-\infty, k]$.

What does the union symbol $\cup$ mean?

The union symbol $\cup$ is used to combine two or more intervals that together form the domain or range. For example, $(-\infty, 2) \cup (2, \infty)$ means all real numbers except 2. The function takes all values from negative infinity up to (but not including) 2, combined with all values from 2 to positive infinity.

How does this calculator handle trigonometric functions?

The calculator supports $\sin(x)$, $\cos(x)$, $\tan(x)$, and their domains/ranges. For $\sin(x)$ and $\cos(x)$, the domain is all real numbers $(-\infty, \infty)$ and the range is $[-1, 1]$. For $\tan(x)$, the domain excludes $x = \frac{\pi}{2} + n\pi$ where the function has vertical asymptotes.

For more math tools, try the Function Grapher, Inequality Solver, and Function Composition Calculator.

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