Domain & Range Finder

Function Setup

Select function type

f(x) = x³ − 2x + 1

Degree (for display)

Description (optional)

All polynomial functions have domain (−∞, ∞). Range depends on degree & leading coefficient.

Examples

f(x) = 1 / (x − 3)

Numerator: ax + b

Denominator: cx + d

Set denominator = 0, solve for excluded x-value(s).

Examples

f(x) = √(x + 4)

Radicand: ax + b (inside √)

a (coefficient of x)

b (constant)

Solve ax + b ≥ 0 to find the domain. Range starts at 0.

Examples

f(x) = |x − 2| + 1

f(x) = |ax + b| + k

k (vertical shift)

Domain is always (−∞, ∞). Range is [k, ∞) since |expr| ≥ 0.

Examples

f(x) = ln(2x − 6)

Argument: ax + b (inside log)

Base (e=natural)

Solve ax + b > 0. Range is always (−∞, ∞).

Examples

f(x) = 3·eˣ + 2

f(x) = A·bˣ + k

A (coefficient)

b (base, e=natural)

k (vertical shift)

Domain is always (−∞, ∞). Range depends on A and k.

Examples

f(x) = { 2x+1 if x < 0; x²+1 if x ≥ 0 }

Piece 1: f(x) = m₁x + b₁ when x < c

m₁

b₁

Piece 2: f(x) = m₂x + b₂ when x ≥ c

m₂

b₂

Breakpoint c

Domain is all reals (−∞,∞). Range is the union of each piece's output.

Examples

Domain & Range

Select a function type, enter values, and press Find Domain & Range to see results in interval notation with step-by-step explanation.

Function

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Domain

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Range

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What Restricts the Domain?

Domain = all x for which f(x) is defined

Polynomials: No restrictions — always (−∞, ∞).

Rational f(x) = p(x)/q(x): Exclude x-values where q(x) = 0 (denominator cannot be zero).

Radical √(expr): Require expr ≥ 0 (can't take the square root of a negative real number).

Logarithm log(expr): Require expr > 0 (log is undefined for zero or negatives).

Absolute value & Exponential: No restrictions — domain is always (−∞, ∞).

Write the domain in interval notation: use [ ] for included endpoints (≤, ≥) and ( ) for excluded endpoints (<, >, ±∞).

How to Find the Range

Range = all possible output values f(x)

Polynomials (odd degree): Range = (−∞, ∞). Even degree: range is bounded below (or above) by the vertex.

Rational (linear/linear): Horizontal asymptote y = a/c is excluded; range is (−∞, a/c) ∪ (a/c, ∞).

Radical √(ax+b): Output ≥ 0, so range = [0, ∞).

Absolute value |ax+b|+k: Output ≥ k, so range = [k, ∞).

Logarithm: Range = (−∞, ∞) — logs can produce any real value.

Exponential A·bˣ+k: A>0 → range (k, ∞); A<0 → range (−∞, k).

Always sketch the graph mentally to confirm your range.
Use the horizontal asymptote to identify excluded range values for rationals.
A vertical shift k moves the entire range up/down by k units.

Domain & range still tricky?