WorkedMathGlossary

Domain and Range Calculator

Pick a function type below to instantly get its domain and range.

Domain
All real numbers
Range
y ≥ 0

Common Domain & Range Rules

  • Linear and quadratic functions: domain is always all real numbers.
  • Square root functions: domain is restricted to where the inside stays non-negative.
  • Reciprocal functions: domain excludes wherever the denominator would be zero.

Worked example

f(x) = √(x − 3) + 2.

The expression under the root needs x − 3 ≥ 0, so x ≥ 3.

Domain: x ≥ 3. Range: y ≥ 2.

A second example

g(x) = 1 / (x − 2) + 5.

The denominator can't be zero, so x − 2 ≠ 0, meaning x ≠ 2.

Domain: x ≠ 2. Range: y ≠ 5 (the fraction term itself can never equal 0).

Keep going

Frequently Asked Questions

What is the domain of a function?

The domain is every input (x-value) the function can accept without breaking a math rule, like dividing by zero or taking the square root of a negative number.

What is the range of a function?

The range is every possible output (y-value) the function can actually produce.

Why does a square root function have a restricted domain?

The square root of a negative number isn't a real number, so a square root function's domain only includes inputs that keep the expression under the radical at zero or above.

What is the domain of a logarithmic function?

A logarithm is only defined for positive inputs, so log(x) has domain x > 0. For a shifted function like log(x − 3), the domain shifts too: x − 3 > 0, so x > 3.

What's a common mistake with domain and range?

Mixing up which is which — domain is always about the inputs (x-values) you're allowed to plug in, and range is about the outputs (y-values) that actually come out. It's easy to accidentally restrict the wrong axis, especially with rational and square root functions.

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