Function Domain & Range Calculator
Determines the domain and range of rational, radical, and logarithmic functions from their coefficients. Use it when analysing function behaviour or preparing graphs in algebra and precalculus.
About this calculator
The domain of a function is the set of all input values x for which the function produces a real output. For a rational function f(x) = (ax + b)/(cx + d), the denominator cannot be zero, so the domain excludes x = −d/c; in interval notation: (−∞, −d/c) ∪ (−d/c, +∞). For a square-root function f(x) = √(ax + b), the radicand must be non-negative: ax + b ≥ 0, giving x ≥ −b/a. Logarithmic functions log(ax + b) require ax + b > 0, so x > −b/a. The range is determined by the output values the function can actually reach — for rational functions with a horizontal asymptote y = a/c, the range excludes that value. Understanding domain and range is essential before plotting, integrating, or composing functions.
How to use
Find the domain of f(x) = (3x + 5)/(2x − 4). Enter function type = Rational, a = 3, b = 5, c = 2, d = −4. Step 1 — set the denominator ≠ 0: 2x − 4 ≠ 0 → x ≠ 2. Step 2 — the formula gives: x ≠ −d/c = −(−4)/2 = 2. Domain: (−∞, 2) ∪ (2, +∞). Step 3 — the horizontal asymptote is y = a/c = 3/2, so the range is (−∞, 3/2) ∪ (3/2, +∞).
Frequently asked questions
How do you find the domain of a rational function step by step?
Write out the denominator expression and set it equal to zero, then solve for x. Those x-values are excluded from the domain because division by zero is undefined. For f(x) = (ax + b)/(cx + d), solve cx + d = 0 to get x = −d/c. Express the domain in interval notation by removing that point from all real numbers. If the denominator is a quadratic, there may be two excluded values or none (when the discriminant is negative).
What is the range of a rational function and how is it different from the domain?
The domain concerns valid inputs; the range concerns reachable outputs. For a simple rational function (ax + b)/(cx + d), the output approaches but never equals the horizontal asymptote y = a/c as x → ±∞. That value is therefore excluded from the range. Unlike polynomials whose range is often all real numbers, rational functions typically have one or more gaps in both domain and range corresponding to their asymptotes. Graphing the function confirms these gaps visually.
Why does a square root function have a restricted domain compared to a polynomial?
A square root (or any even-index radical) requires its radicand to be non-negative in order to produce a real number output. For f(x) = √(ax + b), the condition ax + b ≥ 0 restricts x to a half-line on the number line. Polynomials, by contrast, accept all real x because addition, subtraction, and multiplication never produce undefined results. This restriction on square-root functions also limits their range to non-negative values (y ≥ 0), making both domain and range bounded on one side.