Calculator Input
Example Data Table
| Type | Input A | Input B | Expected Test |
|---|---|---|---|
| Function | f(x) = 2*x + 3 | g(x) = (x - 3) / 2 | f(g(x)) = x and g(f(x)) = x |
| Function | f(x) = x + 7 | g(x) = x - 7 | Both compositions return x |
| Matrix | 1,2; 3,4 | -2,1; 1.5,-0.5 | A × B and B × A return identity |
Formula Used
Function inverse rule: two functions are inverses when f(g(x)) = x and g(f(x)) = x for all valid x values.
Function error: error = |computed composition - x|. The calculator compares this value with your tolerance.
Matrix inverse rule: two square matrices are inverses when A × B = I and B × A = I.
Matrix error: each product cell is compared with the matching identity matrix cell. The largest difference becomes the final error.
How to Use This Calculator
- Choose function inverse or matrix inverse mode.
- Enter the function pair or matrix pair.
- Set a tolerance for rounding and numerical checks.
- For functions, add the sample range and step size.
- Press the submit button to view results above the form.
- Download CSV for spreadsheet use or PDF for reporting.
Why an Inverse Checker Matters
An inverse checker confirms whether two operations undo each other. In maths, this idea appears in functions, matrices, transformations, and problem solving. A function inverse must return the starting input after both compositions are tested. A matrix inverse must create the identity matrix when multiplied in both possible orders. This calculator brings both checks into one practical workspace.
Function Inverse Testing
For functions, the calculator evaluates f(g(x)) and g(f(x)) over a chosen sample interval. Each sample compares the computed result with the original x value. The difference is called the absolute error. A small tolerance is useful because decimal functions may create tiny rounding differences. The tool also records f(x), g(x), both compositions, and pass status for each sample.
Matrix Inverse Testing
For matrices, the checker multiplies A × B and B × A. Both products should match the identity matrix. The identity matrix has ones on the main diagonal and zeros elsewhere. Testing both orders is important because matrix multiplication is not always commutative. The calculator also reports determinant values, product matrices, and the largest identity error found.
Tolerance and Accuracy
The tolerance field controls how strict the verification is. A tolerance of 0.000001 is often suitable for classroom examples. Larger tolerances may help when values come from measurements, rounded decimals, or numerical methods. Smaller tolerances are better for exact algebraic work. The final decision is based on the maximum error across all tested values.
Practical Maths Uses
Students can use this checker to verify algebra homework. Teachers can build examples that show why composition order matters. Engineers can test transformation matrices before using them in models. Data analysts can check linear systems, coordinate conversions, and normalization steps. The CSV export supports spreadsheets. The PDF export gives a quick report for notes, assignments, or review.
Interpreting Results
A passing result means every selected test stayed inside the tolerance. It does not prove every possible value is valid. For a complete proof, domain restrictions and symbolic algebra are still needed. A failing result shows where the inverse relationship breaks. Review the largest error, then check the formula, matrix entries, domain, and rounding choices before trusting the final answer fully today.
FAQs
What does this inverse checker test?
It tests whether two functions or matrices undo each other. Function mode checks both compositions. Matrix mode checks both multiplication orders against the identity matrix.
Does a passing function result prove a true inverse?
No. It gives strong numerical evidence over the selected samples. A full proof still needs domain checks and symbolic reasoning.
Why are both f(g(x)) and g(f(x)) checked?
Both directions matter. A valid inverse relationship should return the original input when the functions are composed in either order.
Why are both A × B and B × A checked?
Matrix multiplication order matters. For inverse matrices, both products must equal the identity matrix within the selected tolerance.
What tolerance should I use?
Use 0.000001 for most decimal examples. Use a smaller value for exact work. Use a larger value for rounded measurements.
Can I use trigonometric functions?
Yes. Supported functions include sin, cos, tan, asin, acos, atan, sqrt, log, ln, exp, and abs. Angles are evaluated in radians.
How should I enter a matrix?
Enter one row per line. Use commas or spaces between values. You may also separate rows with semicolons.
What do CSV and PDF exports include?
The CSV export includes detailed rows for spreadsheet review. The PDF export creates a compact report with the calculated verification table.