Domain Calculator for Multivariable Functions

Calculator

Function expression

Use x, y, z, +, -, *, /, ^, sqrt, log, sin, cos.

Variables

Denominator expressions

Even root radicands

Logarithm arguments

Inverse trig arguments

Manual constraints

Examples: x >= 0, x+y < 7, z != 1.

Sample x

Sample y

Sample z

Grid x minimum

Grid x maximum

Grid y minimum

Grid y maximum

Grid step

Example Data Table

Function	Denominator	Root rule	Log rule	Domain idea
sqrt(x+y)/(z-1)+log(x)	z-1 != 0	x+y >= 0	x > 0	x > 0, x+y >= 0, z != 1
log(9-x^2-y^2)	None	None	9-x^2-y^2 > 0	Inside the circle radius 3
asin((x-y)/5)	None	None	None	-1 <= (x-y)/5 <= 1

Formula Used

The calculator builds an intersection of domain restrictions. It uses this structure:

D = {(x, y, z) in R^n : q(x,y,z) != 0, r(x,y,z) >= 0, l(x,y,z) > 0, -1 <= a(x,y,z) <= 1}

Here q is any denominator. The expression r is an even root radicand. The expression l is a logarithm argument. The expression a is an inverse sine or cosine argument. Manual inequalities are added to the same intersection.

How to Use This Calculator

Enter the multivariable function for display and evaluation.
List denominator expressions, one per line.
List radicands that appear under even roots.
List logarithm arguments that must stay positive.
Add inverse trigonometric arguments and manual constraints.
Enter a sample point and grid range.
Press calculate, then review the result above the form.
Use the CSV or PDF button to save the result.

Advanced domain analysis for several variables

A multivariable domain is the complete set of input points that make a function meaningful. It may use two variables, three variables, or more. Many classroom answers look short, but the checking process can be long. Fractions, square roots, logarithms, and inverse trigonometric terms can each add a separate rule. This calculator keeps those rules visible, so the final domain is easier to review.

Why restrictions matter

A denominator cannot equal zero. An even root needs a nonnegative radicand. A logarithm needs a positive argument. Inverse sine and inverse cosine need an input between negative one and positive one. These rules may overlap. For example, a point can satisfy the logarithm rule but fail the denominator rule. The true domain must satisfy every active rule at the same time.

How the tool studies the domain

The form lets you enter the displayed function and separate restriction lists. This design is useful because symbolic domain work often depends on structure. The calculator evaluates each list at a sample point. It also scans a rectangular grid for x and y while holding z fixed. The scan does not prove the full domain, but it gives a helpful numerical check. It can reveal obvious excluded lines, failed regions, and boundary behavior.

Reading the result

The result begins with a domain-builder statement. It then shows whether the chosen sample point is allowed. Each restriction gets its own pass or fail status. If the sample is valid, the tool also evaluates the function. The grid summary reports tested points, valid points, invalid points, and the valid percentage. Use these values as a check against your algebra.

Best use cases

This page works well for homework checking, lecture examples, and quick reports. It is also useful when a formula mixes several restrictions. Keep one expression per line in each restriction box. Use clear variable names, such as x, y, and z. After solving, download the CSV or PDF result. Save the exported file with your notes, answer key, or worksheet. Always compare the sampled result with exact algebra. Numerical checks guide your work. They do not replace proof. Write final domains with inequalities, exclusions, and interval notation when needed carefully.

FAQs

What is a multivariable domain?

It is the set of all input points that make a multivariable function defined. For f(x,y,z), the domain contains allowed triples. Every restriction must be satisfied.

Can this calculator solve every symbolic domain?

No calculator can safely simplify every symbolic case. This tool builds and tests common restrictions. Use its output with your algebraic reasoning for final proof.

Why enter denominator expressions separately?

Separate denominators make zero-exclusion rules clear. Each denominator must not equal zero. This helps the result table show exactly which rule passed or failed.

What goes in the root box?

Enter expressions under even roots, such as square roots. Each radicand must be greater than or equal to zero for real-valued results.

What goes in the logarithm box?

Enter each log argument only. For log(x-y), enter x-y. The calculator checks that each argument is positive at the sample and grid points.

How are inverse trig restrictions handled?

For asin and acos inputs, the argument must stay between negative one and positive one. Enter only the argument expression for checking.

Does the grid scan prove the domain?

No. The grid scan is a numerical check. It helps detect patterns and failed regions, but exact domain notation still needs algebraic support.

Can I download my result?

Yes. Use the CSV button for spreadsheet data. Use the PDF button for a compact printable summary of the latest calculated result.