Verifying Inverse Functions Calculator

Calculator Inputs

Original function f(x)

Proposed inverse g(x)

Variable name

Manual sample values

Range start

Range end

Generated sample count

Tolerance

Decimal places

Angle mode

Error comparison

Composition check

Example Data Table

f(x)	g(x)	Sample x values	Expected result
(2*x+3)/5	(5*x-3)/2	-2,-1,0,1,2	Both compositions return x.
x^3	x^(1/3)	0,1,8,27	Positive cube samples pass.
exp(x)	ln(x)	1,2,3,4	Use positive domain values.

Formula Used

Two functions are inverses when both compositions return the original input. The calculator checks these conditions numerically:

f(g(x)) = x and g(f(x)) = x

Absolute error is calculated as |composition result - x|. Relative error is calculated as absolute error / max(1, |x|). A row passes when the selected error value is within your tolerance.

How to Use This Calculator

Enter the original function in the f(x) field.
Enter the proposed inverse function in the g(x) field.
Choose manual sample points or leave them blank for a generated range.
Set tolerance, decimal places, angle mode, and comparison type.
Press the verify button and review the result table above the form.
Download the CSV or PDF file for records.

About This Inverse Function Verifier

An inverse function reverses another function. It sends each output back to the original input. This calculator checks that relationship with direct composition tests. It compares f(g(x)) with x. It also compares g(f(x)) with x. When both compositions match, the functions behave like inverses on the tested values.

Why Verification Matters

Many inverse mistakes look correct at first glance. A sign can be wrong. A denominator can hide a restricted value. A square root can require a limited domain. Trigonometric expressions can depend on angle mode. Numeric testing helps reveal these issues quickly. It does not replace a formal proof. It gives strong evidence across the chosen sample set.

Advanced Inputs

The tool accepts common operators, powers, constants, and functions. You can test linear, rational, exponential, logarithmic, radical, and trigonometric forms. You may enter custom sample values. You can also build an evenly spaced test range. The tolerance field controls how close a composition must be to x. Smaller tolerances are stricter. Larger tolerances can help when rounding or complex decimals appear.

Interpreting Results

Each row shows the starting value, the inner result, both compositions, errors, and pass status. The maximum error highlights the weakest tested point. The mean error gives a general accuracy picture. Invalid values are marked, because they may show domain conflicts. A failed row often means the proposed inverse needs algebraic correction.

Practical Use Cases

Students can verify homework answers before writing proofs. Teachers can create examples for inverse function lessons. Developers can test formula transformations used in calculators. Analysts can check model conversions where input and output units move both ways. The export buttons help save evidence for notes, reports, or review sheets.

Best Practice

Choose sample values from the intended domain. Avoid undefined points. Include negative, zero, fractional, and large values when allowed. For restricted functions, test only valid inputs. Then confirm the algebra by solving y = f(x) for x.

Limitations

The calculator uses numerical testing. Different formulas can agree at tested points and fail elsewhere. A final proof should show the compositions symbolically. Still, broad samples improve confidence. Keep the same angle mode for paired trigonometric functions. Review rejected rows before changing the tolerance and sample design choices.

FAQs

What does this calculator verify?

It checks whether two entered functions behave as inverses on selected sample values. It tests f(g(x)) and g(f(x)), then compares each result with x using your tolerance.

Does a numeric pass prove the functions are inverses?

No. A pass gives useful evidence for the tested values. A complete proof still requires algebraic work and correct domain restrictions.

Which operators can I use?

You can use +, -, *, /, ^, parentheses, pi, e, and functions such as sin, cos, tan, sqrt, abs, exp, ln, and log.

How should I choose sample values?

Use values from the intended domain. Include negative numbers, zero, fractions, and larger values when allowed. Avoid undefined inputs.

What tolerance should I enter?

Use a small tolerance like 0.000001 for strict checks. Increase it when decimal rounding creates tiny differences in otherwise correct results.

Why do some rows show invalid values?

Invalid rows usually come from domain problems, division by zero, unsupported operations, or logarithms and roots receiving unsuitable inputs.

Can I test trigonometric inverse functions?

Yes. Select radians or degrees before calculating. Remember that trigonometric inverses often need restricted domains to behave correctly.

What does the CSV export include?

The CSV file includes x, f(x), g(x), both composition results, absolute errors, and row status. Use it for records or further review.