Verification Result
| x | f(x) | g(x) | f(g(x)) | g(f(x)) | f(g(x)) - x | g(f(x)) - x | Status |
|---|
Calculator Input
Example Data Table
| f(x) | g(x) | Sample Points | Expected Result |
|---|---|---|---|
| 2*x + 3 | (x - 3) / 2 | -3, -1, 0, 1, 2 | Inverse pair |
| x^3 | nthRoot(x, 3) | -8, -1, 0, 1, 8 | Inverse pair |
| x^2 | sqrt(x) | 0, 1, 4, 9 | Inverse on restricted domain |
Formula Used
Two functions are inverses when each composition returns the original input.
First test: f(g(x)) = x
Second test: g(f(x)) = x
The calculator evaluates both compositions at selected sample points. It then compares each result with x. A tolerance value controls small rounding differences. If both differences stay inside the tolerance, the functions pass the numeric verification.
Domain and range still matter. For example, f(x) = x² and g(x) = √x are inverse only when f is restricted to x ≥ 0. The tool includes domain and range notes, so your result is clearer.
How to Use This Calculator
Enter the first function in the f(x) field. Enter the possible inverse in the g(x) field. Use x as the variable. Add sample x values separated by commas. Choose a tolerance for decimal comparisons. Add domain and range notes when restrictions are important.
Press the verification button. The result appears above the form and below the header. Review both compositions. Export the table as CSV or PDF when you need a saved report.
Inverse Function Verification Guide
What This Tool Checks
An inverse function reverses another function. If f changes x into y, then the inverse changes y back into x. This calculator checks that idea with composition. It places g(x) inside f(x). It also places f(x) inside g(x). If both expressions return x, the two functions behave as inverses.
Why Composition Matters
Composition gives a direct test. A pair may look related but still fail. Small algebra errors often appear when functions are nested. The table helps reveal those errors. It shows f(x), g(x), f(g(x)), and g(f(x)). It also shows the difference between each composition and the original input.
Using Sample Points Carefully
Sample testing is useful for learning and checking work. It is not a full symbolic proof for every possible input. Still, it can quickly confirm many common problems. Use negative, zero, positive, fraction, and large values when allowed. More varied points give stronger evidence.
Domain and Range Restrictions
Domain and range are essential in inverse problems. Some functions need restrictions before an inverse works. A quadratic function is a common example. Its inverse relation is not a function unless the original domain is limited. The notes fields help you record those limits beside the result.
Understanding Tolerance
Decimal calculations can create tiny rounding differences. A result like 0.999999 may represent 1 in practical work. Tolerance tells the calculator how much difference is acceptable. A smaller tolerance is stricter. A larger tolerance is more forgiving.
Best Use Cases
This calculator supports homework checks, lesson examples, tutoring, and worksheet preparation. It is also helpful for reviewing linear, radical, exponential, logarithmic, and restricted-domain functions. Always combine the numeric result with algebraic reasoning. That gives a stronger and more reliable conclusion.
FAQs
What is an inverse function?
An inverse function reverses the action of another function. If f sends x to y, the inverse sends y back to x.
How do I verify inverse functions?
Check both compositions. If f(g(x)) = x and g(f(x)) = x, the functions are inverses on the stated domain.
Why must both compositions be checked?
Both checks confirm that each function fully reverses the other. This is important when domains and ranges are restricted.
Can sample points prove an inverse?
Sample points provide strong checking support. A full proof usually needs algebra and domain reasoning.
What does tolerance mean?
Tolerance allows small decimal differences. It helps avoid false failures caused by rounding in numeric calculations.
Can I use trigonometric functions?
Yes, math expressions can include supported functions. Be careful with radians, domains, and restricted inverse branches.
Why do domain restrictions matter?
Some functions are not one-to-one everywhere. Restricting the domain can make a valid inverse function possible.
What file exports are available?
You can download the verification table as a CSV file. You can also export a PDF summary report.