Calculator Inputs
This tool supports five invertible families: linear, shifted odd power, exponential, logarithmic, and rational fractional linear.
Formula Used
The calculator follows the standard inverse proof method: write y = f(x), solve algebraically for x, swap x and y, and then verify both compositions: f^-1(f(x)) = x and f(f^-1(x)) = x.
| Family | Original form | Inverse form | Main condition |
|---|---|---|---|
| Linear | f(x) = ax + b | f^-1(x) = (x - b) / a | a ≠ 0 |
| Shifted odd power | f(x) = a(x - h)^n + k | f^-1(x) = h + ((x - k) / a)^(1/n) | a ≠ 0, n odd |
| Exponential | f(x) = a·b^(x - h) + k | f^-1(x) = h + log_b((x - k) / a) | a ≠ 0, b > 0, b ≠ 1 |
| Logarithmic | f(x) = a·log_b(x - h) + k | f^-1(x) = h + b^((x - k) / a) | a ≠ 0, x > h |
| Rational fractional linear | f(x) = (ax + b)/(cx + d) | f^-1(x) = (dx - b)/(a - cx) | ad - bc ≠ 0 |
How to Use This Calculator
- Select the function family that matches the inverse problem you want to prove.
- Enter the relevant coefficients, shifts, exponent, or base.
- Add a test input x to check the identity numerically.
- Set graph bounds so the original curve and its inverse are easy to inspect.
- Press Compute Inverse Proof to generate the inverse, restrictions, proof steps, and graph.
- Use the CSV or PDF buttons to export the computed result summary.
Example Data Table
| Example | Family | Parameters | Original function | Inverse function | Reason it is invertible |
|---|---|---|---|---|---|
| 1 | Linear | a = 2, b = 3 | f(x) = 2x + 3 | f^-1(x) = (x - 3)/2 | Nonzero slope makes the function one-to-one. |
| 2 | Exponential | a = 1, base = 2, h = 0, k = 0 | f(x) = 2^x | f^-1(x) = log_2(x) | Exponential growth is strictly monotonic. |
| 3 | Rational fractional linear | a = 2, b = 1, c = 1, d = 3 | f(x) = (2x + 1)/(x + 3) | f^-1(x) = (3x - 1)/(2 - x) | Determinant ad - bc = 5 is nonzero. |
FAQs
1) What does this calculator prove?
It proves an inverse by solving for x, swapping variables, and checking both compositions. It also reports restrictions, domains, ranges, and a graph.
2) Why must a function be one-to-one?
An inverse must return exactly one output for every valid input. If the original function repeats y-values, the inverse would fail the function test.
3) Why is an odd exponent required in the power family?
An odd power remains one-to-one on all real numbers. Even powers repeat outputs, so they need an added domain restriction before an inverse can exist.
4) Why do the domain and range swap?
Every output of the original becomes an input of the inverse. That is why the original range becomes the inverse domain, and vice versa.
5) Why does the rational family need ad - bc ≠ 0?
That determinant guarantees the algebraic rearrangement works and avoids collapsing the mapping into a non-invertible form. If it is zero, the inverse formula breaks down.
6) How does the graph help verify the inverse?
A correct inverse reflects the original curve across the line y = x. When both graphs mirror each other, the algebraic proof is visually reinforced.
7) What is identity error?
It is the numerical difference after composing the function and inverse using your test input. Small values near zero indicate a successful computational check.
8) Can this calculator handle every possible function?
No. It focuses on five common invertible families. For arbitrary symbolic expressions, a computer algebra system or manual proof may still be necessary.