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.
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.
2) Square roots (even roots): radicand must be nonnegative.
3) Logarithms: argument must be positive.
Common range rule (quadratic vertex form):
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, ∞).
All calculators