Calculator Inputs
Example Data Table
Example pair: f(x) = 2*x + 3 and g(x) = (x - 3)/2.
| x | f(x) | g(x) | f(g(x)) | g(f(x)) |
|---|---|---|---|---|
| -2 | -1 | -2.5 | -2 | -2 |
| 0 | 3 | -1.5 | 0 | 0 |
| 4 | 11 | 0.5 | 4 | 4 |
Formula Used
Two functions are inverses when each undoes the other on a valid shared domain.
Inverse conditions: f(g(x)) = x and g(f(x)) = x
Error measures: E1 = |f(g(x)) - x| and E2 = |g(f(x)) - x|
The calculator samples many x values across the chosen interval, evaluates both compositions, and compares the largest observed errors with the tolerance you set.
How to Use This Calculator
- Enter the first function in the f(x) field.
- Enter the second function in the g(x) field.
- Choose the domain range to test.
- Set the sample count and tolerance.
- Pick the displayed decimal precision.
- Submit the form to run the numerical check.
- Read the result summary above the form.
- Inspect the graph and the table for detail.
- Download the report as CSV or PDF if needed.
About This Calculator
This tool helps you test whether two formulas behave like inverse functions across a chosen interval. Instead of relying on symbolic algebra alone, it samples many x values and evaluates both compositions directly.
That approach is useful when you want a quick practical check for algebra exercises, classroom demonstrations, verification tasks, or function modeling work. It can also reveal where a pair fails because of domain restrictions, undefined values, or rounding effects.
The summary reports the largest composition errors, the number of valid sample points, and whether the observed behavior stays inside your chosen tolerance. The table shows every sample row so you can inspect exact values rather than trusting a single headline result.
The graph adds another perspective. True inverse pairs reflect across the line y = x on intervals where both functions are valid. When the curves do not mirror each other, the composition errors usually confirm the mismatch.
Because this is a numerical checker, the result is best read as strong evidence over the tested range, not a formal proof for every possible input. For full rigor, combine the output with algebraic reasoning about domains, one-to-one behavior, and exact composition simplification.
FAQs
1. What does this calculator verify?
It checks whether two entered functions behave like inverses on a tested interval by evaluating f(g(x)) and g(f(x)) across many sample points and comparing the resulting errors with your chosen tolerance.
2. Why can a true inverse pair show tiny errors?
Computers store many decimals approximately. Trigonometric, logarithmic, and fractional calculations can introduce very small floating-point differences, so exact zero is not always expected in numerical testing.
3. Can I use trigonometric or logarithmic expressions?
Yes. The calculator supports functions such as sin, cos, tan, asin, acos, atan, sqrt, abs, log, log10, exp, pow, floor, ceil, round, and pi for a wide range of math checks.
4. Why do some rows display Invalid?
An invalid row means one step of the evaluation was undefined or outside a valid domain. Common causes include division by zero, square roots of negatives, or logarithms of nonpositive values.
5. Is this result a formal proof?
No. It is a numerical verification over the interval you selected. A formal proof still requires algebraic composition, domain analysis, and confirmation that each function is one-to-one where needed.
6. What input style should I follow?
Use x as the variable and write multiplication explicitly, such as 3*x or x*(x+1). You may enter powers with ^ and group terms with parentheses.
7. Why are both compositions tested?
Both directions matter. A pair should satisfy f(g(x)) = x and g(f(x)) = x on the valid shared domain. Checking only one direction can hide important restrictions or mismatches.
8. What should I look for on the graph?
Look for reflection across the line y = x. When the two curves mirror each other on valid intervals, that visual pattern supports the numerical composition results.