How do I find domain and range with this calculator?

Enter a function of x, such as (2x + 1)/(x - 3), and choose Find domain and range or press Enter. The domain is (-∞, 3) ∪ (3, ∞), and the range is (-∞, 2) ∪ (2, ∞). The restrictions and range working explain the excluded values. After editing either field, submit again to see the current result.

Which functions have a supported range?

Constants, lines, quadratics, fractional-linear functions, and a scaled or shifted square root, logarithm or exponential of a linear expression, with rational coefficients. Supported expressions outside these range families can still receive an exact domain with Range marked Unsupported. For example, x^3 - x has domain (-∞, ∞), but its range is unsupported here.

Does it show the vertex of a quadratic?

Yes. Enter the quadratic and submit it; the Range working gives the vertex and explains whether the parabola opens upward or downward. For x^2 - 4x + 1, the vertex is at x = 2 with f = -3, giving range [-3, ∞) over all real x.

Which domain restrictions can it solve?

Denominators must be nonzero, square-root arguments must be nonnegative, and logarithm arguments must be positive. Polynomial denominator zeros and square-root or logarithm argument restrictions are supported through degree 2; sums, products and supported exponentials can also be read. Use sqrt, ln, log for base 10, log2, log_b, exp or e^. Trigonometric functions, abs, piecewise expressions and arbitrary nested functions are not supported.

Can I graph a function and see its domain and range?

Yes. Enter a supported formula and keep Show sampled plot checked. The page calculates the intervals algebraically, then draws an illustrative plot and labelled number lines. It does not read an uploaded graph picture, a table or a set of points, and the curve samples are never used to guess a range.

What do open and closed interval ends mean?

A square bracket includes the endpoint; a parenthesis excludes it. ∪ joins separate intervals, ∅ means the empty set, and braces mark a single value. Infinite ends are always open. For example, sqrt(x - 2)/(x - 3) includes 2 and excludes 3, giving domain [2, 3) ∪ (3, ∞); its range is unsupported here.

Why does a cancelled denominator still restrict the domain?

Restrictions come from the expression as typed. x/x is undefined at 0, so its domain is (-∞, 0) ∪ (0, ∞) even though it simplifies to 1 wherever defined. Its range is {1}. Cancelling a factor does not restore an excluded input.

How do I restrict the domain?

Use Restrict x to with an interval such as (-2, 1], [0, inf) or (-inf, 3). Leave it blank for all real x. For x^2 on (-2, 1], the domain is (-2, 1] and the range is [0, 4). The result also shows the domain before your interval restriction.

Are irrational endpoints rounded?

No. Algebraic endpoints are shown symbolically, with separately labelled decimal approximations. For sqrt(x^2 - 2), the domain is (-∞, -√2] ∪ [√2, ∞), with √2 ≈ 1.41421. Its range is unsupported here. Decimal approximations do not decide endpoint inclusion or order.

Can I use it free in a browser and save the result?

The core calculator is free without an account or installation, in a current browser on a phone, tablet or computer. Copy intervals copies the result, restrictions and working as text. If clipboard access fails, a selectable text box appears. The last supported function you entered and submitted yourself is saved on this device; opening or loading an example does not replace it.

How do I find domain and range on this scientific calculator?

Use this browser calculator by entering a supported formula in f(x), optionally filling Restrict x to, and choosing Find domain and range. Read the interval results and algebraic working in the result panels. Named calculator models and provider interfaces have their own controls; this page does not give their key sequences.

Domain and Range Calculator

Enter f(x), optionally restrict x to an interval, then choose Find domain and range. Read the domain, supported range, restrictions and endpoint reasons in interval notation. The number lines mark open and closed ends; the sampled function plot is an illustration.

Your input

Example input

f(x), up to 512 charactersRestrict x to (optional)

Optional. Write an interval such as (-2, 1], [0, inf) or (-inf, 3), or leave it blank for all real x. Ends are numbers such as -2, 1.5 or 1/3; inf and -inf are always open.

Show sampled plot (illustration only)

Reads polynomials in x with rational coefficients, their sums, products and quotients when each denominator has degree 2 or less, sqrt and logarithms of polynomials of degree 2 or less, and exponentials of polynomials, such as sqrt(x - 2)/(x - 3), ln(x^2 - 4), 1/sqrt(4 - x^2) and 3·2^(x - 1) + 5. Use x, numbers, + - * / ^ and brackets, with sqrt, ln, log (base 10), log2, log_b(x), exp and e^. 2x means 2 times x, and 1/2x is read as x/2.

Result

Domain(-∞, 3) ∪ (3, ∞)

Range(-∞, 2) ∪ (2, ∞)

Read as f(x) = (2x + 1)/(x - 3).

Domain on a number line

Text equivalent: Number line for the domain: shaded from -∞ to 3 (open dot at 3); then shaded from 3 (open dot at 3) on to ∞. Points are in order and evenly spaced, not to scale.

Range on a number line

Text equivalent: Number line for the range: shaded from -∞ to 2 (open dot at 2); then shaded from 2 (open dot at 2) on to ∞. Points are in order and evenly spaced, not to scale.

Sampled plot, illustration only

Text equivalent: Sampled plot, an illustration only: 362 points of f(x) = (2x + 1)/(x - 3) for x from -3 to 9, with y shown from -19 to 23. Dashed vertical lines mark excluded x: x = 3. Dashed horizontal lines mark values f never reaches: y = 2. The domain and range above are worked out algebraically, not from these samples.

Domain (-∞, 3) ∪ (3, ∞); range (-∞, 2) ∪ (2, ∞).

Domain: (-∞, 3) ∪ (3, ∞)

Range: (-∞, 2) ∪ (2, ∞)

Restriction: x - 3 ≠ 0

Family: fractional-linear (exact range)

Restrictions

From	Condition	Allowed x
Denominator x - 3	x - 3 ≠ 0	(-∞, 3) ∪ (3, ∞)

Endpoints

x = 3 is excluded: the denominator x - 3 is 0 there.

Range

f(x) = (2x + 1)/(x - 3) is fractional-linear: it is decreasing on each side of x = 3 and approaches 2 as x → ±∞ without reaching it.
Solving y = f(x) gives x = (3y + 1)/(y - 2), which has no value at y = 2.
x ∈ (-∞, 3) gives y ∈ (-∞, 2)
x ∈ (3, ∞) gives y ∈ (2, ∞)
y = 2 is excluded: f approaches 2 as x → -∞ but never equals it.

Worked examples

sqrt(x - 2)/(x - 3)radical over rational example
Domain [2, 3) ∪ (3, ∞); range unsupported
x/x, before cancellingcancelled factor example
Domain (-∞, 0) ∪ (0, ∞); simplifies to 1; range {1}
x^2 on (-2, 1]restricted quadratic example
Domain (-2, 1]; range [0, 4)
(2x + 1)/(x - 3)fractional-linear example
Domain (-∞, 3) ∪ (3, ∞); range (-∞, 2) ∪ (2, ∞)
ln(x - 1)logarithm example
Domain (1, ∞); range (-∞, ∞)
sqrt(x^2 - 2)irrational endpoints example
Domain (-∞, -√2] ∪ [√2, ∞); range unsupported

Common questions

How do I find domain and range with this calculator?: Enter a function of x, such as (2x + 1)/(x - 3), and choose Find domain and range or press Enter. The domain is (-∞, 3) ∪ (3, ∞), and the range is (-∞, 2) ∪ (2, ∞). The restrictions and range working explain the excluded values. After editing either field, submit again to see the current result.
Which functions have a supported range?: Constants, lines, quadratics, fractional-linear functions, and a scaled or shifted square root, logarithm or exponential of a linear expression, with rational coefficients. Supported expressions outside these range families can still receive an exact domain with Range marked Unsupported. For example, x^3 - x has domain (-∞, ∞), but its range is unsupported here.
Does it show the vertex of a quadratic?: Yes. Enter the quadratic and submit it; the Range working gives the vertex and explains whether the parabola opens upward or downward. For x^2 - 4x + 1, the vertex is at x = 2 with f = -3, giving range [-3, ∞) over all real x.
Which domain restrictions can it solve?: Denominators must be nonzero, square-root arguments must be nonnegative, and logarithm arguments must be positive. Polynomial denominator zeros and square-root or logarithm argument restrictions are supported through degree 2; sums, products and supported exponentials can also be read. Use sqrt, ln, log for base 10, log2, log_b, exp or e^. Trigonometric functions, abs, piecewise expressions and arbitrary nested functions are not supported.
Can I graph a function and see its domain and range?: Yes. Enter a supported formula and keep Show sampled plot checked. The page calculates the intervals algebraically, then draws an illustrative plot and labelled number lines. It does not read an uploaded graph picture, a table or a set of points, and the curve samples are never used to guess a range.
What do open and closed interval ends mean?: A square bracket includes the endpoint; a parenthesis excludes it. ∪ joins separate intervals, ∅ means the empty set, and braces mark a single value. Infinite ends are always open. For example, sqrt(x - 2)/(x - 3) includes 2 and excludes 3, giving domain [2, 3) ∪ (3, ∞); its range is unsupported here.
Why does a cancelled denominator still restrict the domain?: Restrictions come from the expression as typed. x/x is undefined at 0, so its domain is (-∞, 0) ∪ (0, ∞) even though it simplifies to 1 wherever defined. Its range is {1}. Cancelling a factor does not restore an excluded input.
How do I restrict the domain?: Use Restrict x to with an interval such as (-2, 1], [0, inf) or (-inf, 3). Leave it blank for all real x. For x^2 on (-2, 1], the domain is (-2, 1] and the range is [0, 4). The result also shows the domain before your interval restriction.
Are irrational endpoints rounded?: No. Algebraic endpoints are shown symbolically, with separately labelled decimal approximations. For sqrt(x^2 - 2), the domain is (-∞, -√2] ∪ [√2, ∞), with √2 ≈ 1.41421. Its range is unsupported here. Decimal approximations do not decide endpoint inclusion or order.
Can I use it free in a browser and save the result?: The core calculator is free without an account or installation, in a current browser on a phone, tablet or computer. Copy intervals copies the result, restrictions and working as text. If clipboard access fails, a selectable text box appears. The last supported function you entered and submitted yourself is saved on this device; opening or loading an example does not replace it.
How do I find domain and range on this scientific calculator?: Use this browser calculator by entering a supported formula in f(x), optionally filling Restrict x to, and choosing Find domain and range. Read the interval results and algebraic working in the result panels. Named calculator models and provider interfaces have their own controls; this page does not give their key sequences.

Exact algebraic constraints for supported rational-coefficient expressions; algebraic endpoints are shown symbolically with a labeled decimal approximation. Range is derived analytically only for the listed function families; any other input returns unsupported, never a range read from graph samples. Restrictions are taken from the expression as typed, before any algebraic cancellation.