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Function inputs and outputs

Find a function's domain and range

Enter a supported real-valued function of x. The calculator separates allowed inputs from possible outputs, explains any boundary or excluded value, and shows the result in interval notation.

Use the math keyboard or your device keyboard.
Try an example

Result

Domain starts at 2; range starts at 0

Dom⁡(f)=[2,∞)Ran⁡(f)=[0,∞)\begin{aligned}\operatorname{Dom}(f)&=[2,\infty)\\\operatorname{Ran}(f)&=[0,\infty)\end{aligned}
Conditions
  • Real-valued square roots require their radicand to be at least zero.
  • The endpoint x=2 is included because √0=0 is defined.
  • The displayed graph is an illustration; the interval conclusion follows from the algebraic condition.

Steps

  1. Require a real radicand Inputs are allowed only when the quantity inside the square root is not negative.x−2≥0x-2\ge0
  2. Solve the input condition Adding 2 gives the inclusive lower bound for the domain.x≥2⇒Dom⁡(f)=[2,∞)x\ge2\quad\Rightarrow\quad\operatorname{Dom}(f)=[2,\infty)
  3. Find possible outputs The principal square root equals 0 at x=2 and has no finite upper bound.f(2)=0,f(x)≥0⇒Ran⁡(f)=[0,∞)f(2)=0,\quad f(x)\ge0\quad\Rightarrow\quad\operatorname{Ran}(f)=[0,\infty)
Independent check

At the included boundary, f(2)=0. Below it, x=1 gives √(−1), not a real output; for any y≥0, x=y²+2 is an allowed input with f(x)=y.

Scope

What this domain and range covers

The domain is every real input x for which a function is defined; the range is every real output it can attain. For f(x)=√(x−2), the radicand must be nonnegative, so x≥2. The square root then starts at 0 and grows, giving domain [2,∞) and range [0,∞).

Linear, constant, and quadratic functions

For a constant, report its one-value range. For a nonconstant line, both domain and range are all real numbers. For a quadratic, use the vertex and opening direction to find the attained minimum or maximum.

Examples: 2x−5, 4, x²−6x+8

Square roots of linear expressions

Solve the nonnegative-radicand condition. The principal square root includes 0 when the radicand can equal 0 and never gives a negative real output.

Examples: √(x−2), √(5−2x)

Natural logarithms of linear expressions

Require a strictly positive argument. The zero boundary is excluded, while a nonconstant linear argument can produce every real logarithm output over its valid side.

Examples: ln(x+3), ln(2x−1)

Nonzero constant over a linear denominator

Exclude the denominator's zero from inputs. A nonzero numerator divided by a finite nonzero denominator cannot produce zero as an output.

Examples: 1/(x−4), −2/(3x+6)

How to use it

Enter enough information for one clear task

  1. 1
    Enter one supported expression

    Use x as the variable. A polynomial of degree at most two, √(linear), ln(linear), or a nonzero constant divided by a linear expression can be analyzed in this release.

  2. 2
    Read domain and range separately

    Domain answers which x-values are allowed; range answers which y-values are attained. Brackets include a finite endpoint, while parentheses exclude it or accompany infinity.

  3. 3
    Check the reason for an endpoint

    For a square root, test the zero-radicand input. For a logarithm or reciprocal, test the excluded boundary. For a quadratic, substitute the vertex x-coordinate.

Worked inputs

Examples to try

Use these examples to recognize the method, compare equivalent forms, and check your own work.

A line reaches every real output

No real input is excluded, and solving y=2x−5 gives an x for every real y.

f(x)=2x−5f(x)=2x-5

Expected result

Dom⁡(f)=R,Ran⁡(f)=R\operatorname{Dom}(f)=\mathbb R,\quad\operatorname{Ran}(f)=\mathbb R

A constant has a one-value range

Every real x is allowed, but each input produces exactly 4.

f(x)=4f(x)=4

Expected result

Dom⁡(f)=R,Ran⁡(f)={4}\operatorname{Dom}(f)=\mathbb R,\quad\operatorname{Ran}(f)=\{4\}

A quadratic with a minimum

Complete the square to get (x−3)²−1; its least value is −1.

f(x)=x2−6x+8f(x)=x^2-6x+8

Expected result

Dom⁡(f)=R,Ran⁡(f)=[−1,∞)\operatorname{Dom}(f)=\mathbb R,\quad\operatorname{Ran}(f)=[-1,\infty)

A downward quadratic has a maximum

The negative square is at most zero, with its maximum at x=−1.

g(x)=−(x+1)2+4g(x)=-(x+1)^2+4

Expected result

Dom⁡(g)=R,Ran⁡(g)=(−∞,4]\operatorname{Dom}(g)=\mathbb R,\quad\operatorname{Ran}(g)=(-\infty,4]

A square-root endpoint is included

Require x−2≥0; at x=2, the output 0 exists.

f(x)=x−2f(x)=\sqrt{x-2}

Expected result

Dom⁡(f)=[2,∞),Ran⁡(f)=[0,∞)\operatorname{Dom}(f)=[2,\infty),\quad\operatorname{Ran}(f)=[0,\infty)

A reversed square-root inequality

Solve 5−2x≥0, reversing the inequality when dividing by −2.

h(x)=5−2xh(x)=\sqrt{5-2x}

Expected result

Dom⁡(h)=(−∞,52],Ran⁡(h)=[0,∞)\operatorname{Dom}(h)=(-\infty,\tfrac52],\quad\operatorname{Ran}(h)=[0,\infty)

A logarithm excludes its boundary

Require x+3>0. A logarithm of a positive argument can attain any real output.

f(x)=ln⁡(x+3)f(x)=\ln(x+3)

Expected result

Dom⁡(f)=(−3,∞),Ran⁡(f)=R\operatorname{Dom}(f)=(-3,\infty),\quad\operatorname{Ran}(f)=\mathbb R

A reciprocal excludes one input and one output

The denominator cannot be zero, and a nonzero numerator cannot yield zero.

r(x)=1x−4r(x)=\frac1{x-4}

Expected result

Dom⁡(r)=R∖{4},Ran⁡(r)=R∖{0}\operatorname{Dom}(r)=\mathbb R\setminus\{4\},\quad\operatorname{Ran}(r)=\mathbb R\setminus\{0\}

Complete example

Find both intervals for a decreasing square root

The negative coefficient on x changes which side of the boundary is allowed. The principal square root still cannot be negative.

h(x)=5−2xh(x)=\sqrt{5-2x}
  1. 1
    Find the boundary

    Set the radicand to zero to locate the finite endpoint.

    5−2x=0⇒x=525-2x=0\quad\Rightarrow\quad x=\frac52
  2. 2
    Solve the permitted-input inequality

    Dividing by negative 2 reverses the inequality sign.

    5−2x≥0⇒x≤525-2x\ge0\quad\Rightarrow\quad x\le\frac52
  3. 3
    Use the smallest output and growth direction

    At the boundary the output is 0; as x moves left, the radicand grows without bound.

    h ⁣(52)=0,h(x)≥0h\!\left(\frac52\right)=0,\qquad h(x)\ge0
Dom⁡(h)=(−∞,52],Ran⁡(h)=[0,∞)\operatorname{Dom}(h)=(-\infty,\tfrac52],\quad\operatorname{Ran}(h)=[0,\infty)

Verification: At x=5/2, h(x)=0; x=3 is excluded because the radicand is −1. Every y≥0 is reached by x=(5−y²)/2, which also satisfies x≤5/2.

Avoidable errors

Common mistakes and how to fix them

Using domain as range

Problem: List x≥2 as both the allowed inputs and the possible outputs of √(x−2).

Why it matters: The input begins at 2, but the function value at that input is 0.

Better approach: Answer the two questions separately: which x are permitted, and which y can the function actually produce?

Including a forbidden endpoint

Problem: Use a square bracket at x=−3 for ln(x+3).

Why it matters: At that x the logarithm would receive 0, which has no real logarithm.

Better approach: Use an open parenthesis at strict boundaries or points excluded by division by zero.

Reading the plot as a proof

Problem: Assume a sampled graph proves that an asymptote or a narrow gap is absent.

Why it matters: Finite plot samples can miss forbidden inputs, exact endpoints, and values approached but never reached.

Better approach: Derive conditions from the expression; use the graph only to check the shape and boundary behavior.

Forgetting a negative inequality reversal

Problem: Solve 5−2x≥0 as x≥5/2.

Why it matters: Dividing an inequality by −2 reverses its direction.

Better approach: Test a value on each side of 5/2 or explicitly flip the inequality sign.

Trust the result for the right reasons

Checks, assumptions, and limits

How results are checked

  • A candidate expression is first classified into a supported algebraic family; unsupported compositions are not given a guessed interval.
  • Endpoint inclusion is derived from the exact square-root, logarithm, denominator, or vertex condition.
  • A boundary value and one point on each relevant side are checked where the expression is defined.
  • The graph is sampled only on valid real inputs and cannot override the symbolic interval result.

When to stop and revise the input

  • The calculator handles real-valued constants, linear and quadratic polynomials, square roots or natural logarithms of nonconstant linear expressions, and a nonzero constant over a nonconstant linear denominator.
  • Arbitrary compositions, piecewise functions, trigonometric functions, higher-degree polynomials, and expressions with several interacting restrictions are outside this release.
  • The default domain is the maximal real domain of the expression. A word problem may impose additional real-world restrictions not present in its formula.
  • A graph is an illustration, not a proof of an exact endpoint or a missing output.

Common questions

Find a function's domain and range FAQ

What is the difference between domain and range?

The domain contains allowed input x-values; the range contains output y-values the function actually attains. They can have different endpoints even for the same function.

Why are some endpoints in brackets and others in parentheses?

A bracket means the finite endpoint is included. A parenthesis means it is excluded; infinity always uses a parenthesis because it is not a real endpoint.

Does a graph give the exact domain and range?

A graph helps visualize behavior, but plotted samples cannot establish an exact hole or endpoint. The calculator uses the function's algebraic restrictions and checks the plot against that result.

Why is zero excluded from the range of 1/(x−4)?

A nonzero numerator divided by a finite nonzero number cannot equal zero. The graph approaches y=0 but never touches it.

Can this tool solve every function?

No. It covers the listed elementary real-valued families and rejects unsupported forms instead of inventing a domain or range. A piecewise function or a nested composition needs separate analysis.

Sources and curriculum alignment

This page follows standard introductory mathematics notation and learning sequences. Use these references to continue with a complete course treatment.

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Created by Mathos AI. Methods, conditions, and checks are shown so you can review the mathematical reasoning.