Domain Algebra Function Calculator

Calculator Inputs

Function f(x)

Use x, +, -, *, /, ^, sqrt(), log(), sin(), cos(), abs().

Minimum x

Maximum x

Scan steps

Angle note

Server functions evaluate trigonometry in radians.

Output precision

Example Data Table

Example Function	Main Restriction	Expected Domain
`1/(x-3)`	Denominator cannot be zero.	All real x except 3.
`sqrt(x-2)`	Radicand must be at least zero.	[2, ∞)
`log(x+4)`	Log argument must be positive.	(-4, ∞)
`sqrt(9-x^2)`	Radicand must be nonnegative.	[-3, 3]
`(x+1)/(x^2-9)`	Denominator roots are excluded.	All real x except -3 and 3.

Formula Used

The calculator checks the real-valued domain with algebra rules and numerical scanning.

Polynomial rule: polynomial expressions usually accept all real values.
Rational rule: if f(x)=p(x)/q(x), then q(x) ≠ 0.
Square-root rule: if f(x)=sqrt(g(x)), then g(x) ≥ 0.
Log rule: if f(x)=log(g(x)), then g(x) > 0.
Inverse trig rule: if asin(g(x)) or acos(g(x)), then -1 ≤ g(x) ≤ 1.

The interval result is approximate inside the selected scan window. Increase scan steps for tighter numerical checking.

How to Use This Calculator

Enter an algebraic expression using x as the variable.
Set the minimum and maximum x values for the scan window.
Increase scan steps when the function has sharp changes.
Press the calculate button to show the result above the form.
Review the domain interval, exclusions, graph, and rule notes.
Use the CSV or PDF button to save your result.

Understanding Domain Algebra

What a Domain Means

A function domain is the full set of input values that keep a rule valid. In algebra, this usually means checking every operation inside the expression. Some operations accept all real numbers. Others create limits. Division, radicals, logarithms, and inverse trigonometric functions are common sources of restrictions.

Why a Calculator Helps

A domain calculator helps students inspect those restrictions faster. It also supports a stronger habit. You still read the rule. Then you compare the computed result with known algebra rules. This page scans the selected x window. It marks valid points and invalid points. It also searches for denominator roots that create holes or vertical exclusions.

Rational and Radical Rules

Rational expressions need special care. A denominator cannot equal zero. For example, 1 divided by x minus 3 excludes x equals 3. The graph may approach a vertical asymptote there. The function may still be defined on both sides. The domain writes that missing value clearly.

Radical expressions also need attention. For real outputs, an even root needs a nonnegative radicand. Square root of x minus 2 starts at x equals 2. Values below 2 are not real. Odd roots are less restrictive, but this calculator focuses on common real algebra forms.

Log Rules and Graph Checks

Logarithmic functions have another rule. Their argument must be positive. Log of x minus 4 requires x greater than 4. The boundary is excluded. That is why interval brackets matter. Parentheses show excluded endpoints. Square brackets show included endpoints.

Graphs are useful because they reveal gaps. However, a graph is not a complete proof. A thin hole can hide between sample points. Use the detected restrictions section with the formula notes. Increase the scan samples when an expression changes quickly. Enter wider limits when you want a broader view.

Study Use

This calculator is built for learning and checking. It shows a numerical domain inside your chosen window. It reports approximate range, valid point counts, and possible excluded x values. Export options help you save classroom work. The example table gives ready functions for practice. Always confirm final answers with algebraic reasoning.

For advanced checks, test simplified forms too. Cancelled factors may change graph shape, yet the original expression still keeps its excluded input values for accuracy.

FAQs

What is the domain of a function?

The domain is the set of input values that make the function valid. For real algebra, it excludes values causing division by zero, negative square roots, invalid logarithms, or restricted inverse trigonometric inputs.

Can this calculator prove the exact domain?

It gives a strong numerical check and lists common algebra rules. Exact proof still needs symbolic reasoning, especially for complex expressions, removable holes, or restrictions hidden by simplification.

Why do I need minimum and maximum x values?

The graph and scanner work inside a selected window. Wider limits show more behavior. Smaller limits help inspect a specific interval with better visual detail.

Why are denominator zeros excluded?

Division by zero is undefined. If a denominator becomes zero at a value of x, that input cannot be in the domain, even when the graph looks close to a curve there.

How are square roots handled?

For real-valued results, the value inside a square root must be zero or positive. Negative radicands are marked invalid during the scan.

How are logarithms handled?

The argument inside log, ln, or log10 must be positive. Zero and negative arguments are excluded from the real domain.

Why can a graph miss a hole?

A removable hole may be very narrow. If the scan steps do not land near it, the graph can look continuous. Always compare the result with denominator factors.

Which functions are supported?

You can use x, powers, arithmetic, sqrt, log, ln, log10, abs, exp, common trigonometric functions, and constants pi and e. Use parentheses for clear input.