Domain and Range Calculator

Find valid inputs and possible outputs from a function formula, with restrictions written in interval or set-builder notation.

Drag & drop or upload an image or PDF

Enter a function, such as f(x) = sqrt(x - 3) + 2

Enter the full formula and state any given domain. This tool analyzes formulas, not an unlabeled graph image.

Try an example

Domain describes inputs and range describes outputs

The domain is the set of inputs for which a function is defined. A formula can exclude inputs through division by zero, even roots of negative values, logarithms of nonpositive values, or an explicit restriction.

The range is the set of outputs the function can actually produce. Finding it may require solving for the input, locating a maximum or minimum, using known parent-function ranges, and checking whether boundary or asymptotic values are attained.

How to use the domain and range calculator

Enter the problem as written

Type or paste the full expression. You can also upload a clear photo or PDF and check the extracted text before solving.

Read the working, not only the answer

Each transformation is separated and explained so you can compare it with your own method.

Ask about any step

Continue in the same solution to request another method, check a restriction, or ask why a rule applies.

Domain and range as sets

The domain filters valid inputs. The range collects every output produced by those inputs.

Dom (f) = {x \in R : f (x) is defined}, Range (f) = {f (x) : x \in Dom (f)}

A reliable way to work through it

Find every input restriction

Check denominators, even roots, logarithms, piecewise conditions, and any domain stated with the function.

Analyze possible outputs

Use transformations, extrema, asymptotes, or solve x in terms of y to determine which outputs occur.

Choose endpoint notation carefully

Use a bracket only when a finite endpoint is included. Infinity always takes a parenthesis.

Worked example

Find the domain and range of f(x) = sqrt(x - 3) + 2

x - 3 \geq 0

A real square root requires a nonnegative radicand.

x \geq 3 ⟹ Dom (f) = [3, \infty)

Solve the domain restriction and write it in interval notation.

x - 3 \geq 0

The principal square root cannot produce a negative value.

f (x) = x - 3 + 2 \geq 2 ⟹ Range (f) = [2, \infty)

The vertical shift raises the smallest output from 0 to 2.

The domain is [3, infinity) and the range is [2, infinity).

Common mistakes to check

Checking only the denominator

Radicals, logarithms, inverse functions, and piecewise conditions can also restrict the domain.

Treating an asymptote as included

A function may approach a value indefinitely without ever producing that output.

Using brackets with infinity

Infinity is not a reachable endpoint, so interval notation always uses a parenthesis beside it.

Inverse Function CalculatorReverse a one-to-one function, track domain restrictions, and verify the result by composition.Function Composition CalculatorSubstitute one function into another in the correct order, simplify it, and determine where the composition is defined.Rational Expression CalculatorSimplify algebraic fractions and keep the original domain restrictions with the answer.Logarithm CalculatorEvaluate, expand, condense, or solve logarithms while keeping every domain restriction visible.

Questions students ask

What is the difference between domain and range?

Domain is the set of allowed inputs. Range is the set of outputs produced from those inputs.

Why are denominator zeros excluded?

A function is undefined where its formula requires division by zero, even if later algebra appears to cancel the factor.

Can the answer use interval notation?

Yes. The result can include interval notation and an equivalent set-builder condition.

Can this find domain and range from a graph?

This page is designed for function formulas. A clearly labeled uploaded problem may help, but an unlabeled graph should not be treated as reliable formula input.