Inverse Function Input Panel
Example Data Table
| Function Type | Values | Original Function | Inverse Function | Useful Restriction |
|---|---|---|---|---|
| Linear | a=2, b=3 | f(x)=2x+3 | f⁻¹(y)=(y-3)/2 | None |
| Quadratic | a=1, h=2, k=4 | f(x)=(x-2)²+4 | f⁻¹(y)=2+√(y-4) | Use x ≥ 2 |
| Rational | a=2, b=3, c=1, d=4 | f(x)=(2x+3)/(x+4) | f⁻¹(y)=(3-4y)/(y-2) | y ≠ 2 |
| Exponential | a=2, b=3, h=1, k=5 | f(x)=2·3^(x-1)+5 | f⁻¹(y)=1+ln((y-5)/2)/ln(3) | y > 5 |
Formula Used
Core Inverse Rule
Start with y = f(x). Solve the equation for x. The solved value becomes f⁻¹(y).
y = ax + b
f⁻¹(y) = (y - b) / a
y = a(x - h)² + k
f⁻¹(y) = h ± √((y - k) / a)
y = (ax + b) / (cx + d)
f⁻¹(y) = (b - dy) / (cy - a)
y = a · b^(x - h) + k
f⁻¹(y) = h + ln((y - k) / a) / ln(b)
How to Use This Calculator
- Select the function family that matches your equation.
- Enter the needed constants. Unused fields can stay unchanged.
- Choose a left or right branch for non-one-to-one shapes.
- Enter a test x value and a test y value.
- Press calculate to see the inverse, checks, graph, and table.
- Use CSV or PDF buttons to save the result.
Inverse Functions Explained
What an Inverse Means
An inverse function reverses the action of the original function. If a function changes x into y, the inverse changes that y back into x. This is why inverse notation uses f⁻¹. It does not mean a reciprocal. It means a reverse mapping.
Why Domains Matter
Many functions are not naturally one-to-one. A quadratic is the common example. The same y value can come from two different x values. That creates two possible inverse answers. A branch restriction solves this problem. The calculator lets you pick a left branch or a right branch.
How the Calculator Checks Work
The page does more than show a formula. It also verifies the answer. After finding f⁻¹(y), it places that result back inside the original function. If f(f⁻¹(y)) returns the chosen y, the inverse is working. A small error can appear because decimals are rounded.
Supported Function Families
This calculator supports linear, quadratic, rational, exponential, logarithmic, and power functions. These cover many classroom and applied algebra problems. Each family uses a different inverse rule. The formula section shows the rule used for each case. This helps users learn the method, not only copy the result.
Graph and Table Use
The graph compares the original function and its inverse. It helps reveal symmetry and restrictions. The generated table gives sample y values, inverse x values, and verification values. This is useful for homework checking, lesson planning, and quick numerical review.
Best Practice
Always check the domain before trusting an inverse. Confirm that restricted functions use the correct branch. For exponential and logarithmic forms, respect range and domain limits. For rational forms, avoid denominator restrictions. These checks prevent most inverse function mistakes.
FAQs
1. What is an inverse function?
An inverse function reverses the original function. If f changes x into y, then f⁻¹ changes y back into x when the inverse exists.
2. Why do some functions need a branch?
Some functions give the same output for more than one input. A branch restricts the domain, so each output has one matching input.
3. Is f⁻¹ the same as 1/f?
No. f⁻¹ means inverse function. It is not the reciprocal unless a special problem clearly defines it that way.
4. How is the inverse verified?
The calculator places the inverse result back into the original function. If f(f⁻¹(y)) equals y, the inverse passes the check.
5. Why can the error be a tiny decimal?
Computers use decimal approximations for roots, logs, and powers. Very small errors usually come from rounding, not wrong algebra.
6. Which functions are supported?
The calculator supports linear, quadratic, rational, exponential, logarithmic, and power forms. Each type has its own inverse formula.
7. Why is my y value rejected?
Your y value may be outside the valid range. For example, some quadratic and exponential functions cannot produce every real y value.
8. Can I export the calculation?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a simple report with formulas and sample checks.