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Domain and Range Calculator

Find the domain and range of common functions with clear steps. Supports polynomials, rational functions, square roots, absolute value, basic logarithms, and piecewise rules — plus a mini number-line visual.

Background

Domain is every x you’re allowed to plug in (no division by zero, no negative inside even roots, no log of nonpositive). Range is every y the function can output. Some ranges are “all real numbers,” while others are restricted by graphs (like quadratics) or by transformations.

Enter inputs

Tip: If your range depends on solving a messy equation, use Domain-only mode (still shows a great number-line).

Use x as the variable. Use parentheses, and write powers like x^2. For abs you can use | | or abs().

For log: use log() or ln(). Example: ln(2x+3).

Options

Chips fill values and solve immediately.

Result:

No results yet — enter inputs and click Calculate.

How to use this calculator

  • Choose a function type and enter your expression in x.
  • Domain rules: no division by 0, no negative inside √ (even roots), log inputs must be > 0.
  • Range: works best for quadratics, | |, √, and log transformations.
  • Use Quick picks to see common patterns fast.

How this calculator works

  • It scans your expression for domain restrictions: denominators, sqrt() radicands, and log() arguments.
  • It solves the restriction inequalities for common patterns and formats the answer in interval notation.
  • For range, it uses known range rules for quadratics, | |, √, and log (plus vertical shifts).

Formula & Equation Used

Domain restrictions:

1) Rational functions: denominator cannot be zero.

den(x) ≠ 0

2) Square roots (even roots): radicand must be nonnegative.

radicand ≥ 0

3) Logarithms: argument must be positive.

argument > 0

Common range rule (quadratic vertex form):

y=a(x−h))2+k

If a > 0, then y ≥ k. If a < 0, then y ≤ k.

Example Problem & Step-by-Step Solution

Example 1 — Rational

For (x+1)/(x-3), the denominator can’t be 0 → x ≠ 3.

Example 2 — Square root

For sqrt(5-x), you need 5-x ≥ 0 → x ≤ 5.

Example 3 — Log

For ln(2x+3), you need 2x+3 > 0 → x > -3/2.

Frequently Asked Questions

Q: What’s the difference between domain and range?

The domain is the set of allowed input values (x). The range is the set of possible output values (y).

Q: Why does √ require the radicand ≥ 0?

For real-number outputs, the square root of a negative number is not real, so the inside of √ must be ≥ 0.

Q: Why is x ≠ a for rational functions?

Because division by zero is undefined. Any x-value that makes the denominator 0 must be excluded from the domain.

Q: Does ln(x) allow x = 0?

No. For ln(x) and log(x), the input must be > 0. So x = 0 is not allowed.

Q: How do I write answers in interval notation?

Use parentheses for excluded endpoints and brackets for included endpoints. Example: (-∞, -2] ∪ [2, ∞).