Domain and Range Calculator

Math Algebra • Functions

Written by STEM Calculators Team Published December 13, 2025 Updated February 24, 2026

Press Calculate to estimate the domain and range on your chosen window.

Tip: to explore the full domain, widen the analysis window (increase \(x_{\min}\) negatively and/or increase \(x_{\max}\) positively). For exact symbolic solving of constraints like \(x^2-4\ge 0\), jump to the Inequalities chapter.

Drag to pan. Mouse wheel to zoom (cursor-centered). Double-click to reset view. Probe point shows whether \(x\) is in the domain and its \(f(x)\) value (if defined).

Number line: thick green segments are domain intervals (within the chosen window).

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3. Domain And Range Calculator

The domain of a function is the set of input values for which the expression is real and defined. The range is the set of output values the function can attain. This calculator combines rule-based restriction detection with numerical sampling on a chosen window.

1) Common domain restrictions

Fractions: if \(f(x)=\dfrac{p(x)}{q(x)}\), then require \(q(x)\ne 0\).
Square roots: if \(f(x)=\sqrt{g(x)}\), then require \(g(x)\ge 0\).
Logarithms: if \(f(x)=\ln(g(x))\) or \(\log(g(x))\), then require \(g(x)>0\).
Powers: \(g(x)^{r}\) is real for negative \(g(x)\) only in special cases (e.g. integer \(r\)). For general non-integer exponents, require \(g(x)>0\).

2) How the calculator approximates domain & range

On the chosen analysis window \([x_{\min},x_{\max}]\), the tool samples many points and checks whether the function evaluates to a real finite number. Valid runs become domain intervals, and boundaries are refined by a bisection-like search near the valid/invalid transition.

The range is approximated by taking the minimum and maximum of valid sampled outputs in the window (or by Monte-Carlo sampling in 2D mode).

3) Infinity inputs in practice

You can type infinity as inf, infty, infinity, or ∞ in the window bounds. Since graphs require a finite window, the calculator uses an \(\infty\) proxy value for plotting and sampling. For example, \(x_{\min}=-\infty\) becomes \(-20\) when the proxy is 20.

4) Example

\[ f(x)=\frac{\sqrt{x^2-4}}{x+1}. \]

\(\sqrt{x^2-4}\) requires \(x^2-4\ge 0 \Rightarrow x\le -2 \text{ or } x\ge 2\).
\(x+1\ne 0 \Rightarrow x\ne -1\).

Combine restrictions to get the domain (exact): \(x\in(-\infty,-2]\cup[2,\infty)\) excluding \(x=-1\) (which is already outside those intervals). The calculator will approximate this behavior in your chosen window.

Frequently Asked Questions

What does domain mean for a function f(x)?

The domain is the set of x-values for which f(x) is defined as a real, finite number. Points that cause division by zero or non-real values (like sqrt of a negative number) are excluded.

How does the calculator estimate the range?

It samples many valid x-values in the chosen window and records the corresponding outputs. The range is approximated from the smallest and largest sampled outputs that are real and finite.

Why do I need to choose xmin and xmax?

The tool estimates domain and range numerically on a specific interval, so the window controls what part of the function is explored. Widening the window can reveal additional valid intervals or larger output values.

What restrictions are checked for real-valued functions?

Common checks include denominators not equal to 0, sqrt inputs required to be at least 0, and ln/log inputs required to be greater than 0. Some powers can also force additional restrictions when non-integer exponents are involved.

How is infinity handled in the window bounds?

Infinity entries are treated as very large finite proxy bounds for sampling and plotting. This allows exploration of wide intervals while keeping the computations and graph display finite.

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