Domain and Range Calculator
Find the domain, range, key restriction, and interval notation for common algebra functions, with explanations that match classroom methods.
Domain and range are not the same question
The domain asks which x-values are allowed. The range asks which y-values can actually come out. OpenStax introduces domain and range with equation-based and graph-based methods in its domain and range section, including restrictions from denominators and square roots.
For calculator pages, this distinction matters because a formula can look simple but still have hidden restrictions. A denominator cannot be zero, a square root cannot have a negative radicand in real-number algebra, and a quadratic has a highest or lowest y-value at its vertex.
Quick rules by function type
| Function type | Domain clue | Range clue |
|---|---|---|
| Polynomial | Usually all real numbers | Quadratics use the vertex |
| Rational | Exclude denominator zeros | Check horizontal asymptote/holes |
| Square root | Radicand must be ≥ 0 | Starts at the shifted endpoint |
Common mistake: using only the graph shape
Graphs help, but they can hide exact endpoints. For example, √(x - 3) starts at x = 3, so the domain is [3, ∞), not merely “x is positive.” A rational graph might approach a horizontal asymptote without ever touching it. A good answer explains the restriction, not just the final interval notation.
Common questions
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The domain is the set of all input values that the function is allowed to use. In basic algebra, you usually look for values that would cause division by zero, a negative number under an even root, or another undefined operation. If no restriction appears, the domain may be all real numbers.
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The range is the set of all output values the function can produce. Finding the range often requires more thinking than finding the domain. For a quadratic, the vertex gives the minimum or maximum. For a square root function, the endpoint and vertical direction determine the range. For a rational function, asymptotes and holes may matter.
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A denominator cannot equal zero because division by zero is undefined. If a value of x makes the denominator zero, that x-value is excluded from the domain even if the expression can later be simplified. The original function decides the original domain.
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Use parentheses for endpoints that are not included and brackets for endpoints that are included. For example, x > 2 is written (2, ∞), while x ≥ 2 is written [2, ∞). If one value is excluded from all real numbers, split the interval, such as (-∞, 3) ∪ (3, ∞).
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Yes. Linear functions with nonzero slope have range all real numbers, and many odd-degree polynomials do too. But quadratics, square root functions, and many rational functions have limited ranges. You should decide from the function type, not from guessing.