Calculator
Example Data Table
| Function | Target y | Expected inverse value | Reason |
|---|---|---|---|
| 2*x+3 | 11 | 4 | Subtract 3, then divide by 2. |
| x^2 | 25 | -5 and 5 | Two branches exist without domain restriction. |
| sqrt(x+4) | 3 | 5 | Square 3, then subtract 4. |
| ln(x) | 2 | 7.389056 | Use the exponential form. |
Formula Used
The inverse of a function reverses the mapping made by the original function. If f(x) = y, then f⁻¹(y) = x.
To find a symbolic inverse, write y = f(x). Then solve the equation for x. Finally, replace x with f⁻¹(y).
For a linear function f(x) = ax + b, the inverse is f⁻¹(y) = (y - b) / a.
For a rational function f(x) = (ax + b) / (cx + d), the inverse is f⁻¹(y) = (b - dy) / (cy - a).
The numeric method solves f(x) - y = 0 over the selected interval. It scans for sign changes and refines roots by bisection.
How to Use This Calculator
- Enter a function using x as the variable.
- Use * for multiplication, such as 3*x+2.
- Enter the y value that needs an inverse x value.
- Set a domain interval for the numeric search.
- Select all roots or a single branch.
- Press the calculate button.
- Review the inverse, range estimate, and table.
- Download the CSV or PDF report if needed.
Inverse Function Calculator Guide
What This Tool Does
An inverse function calculator helps you reverse a function. It starts with a value of y and searches for the matching value of x. This is useful in algebra, precalculus, graph analysis, and exam practice. Many functions have simple inverses. Others need a domain restriction before the inverse becomes a true function. This calculator supports both ideas. It checks symbolic patterns first. Then it uses a numeric search over your chosen interval.
Why Domain Matters
A function must be one-to-one before its inverse can also be a function. A straight line with nonzero slope is one-to-one on its whole domain. A parabola is not one-to-one on all real numbers. For example, x² gives the same y value for positive and negative x values. You can fix this by restricting the domain. Choose the lower or upper branch when a function produces more than one possible answer.
Symbolic and Numeric Work
The calculator tries common symbolic forms, including linear, rational, power, radical, exponential, logarithmic, and quadratic expressions. When a direct template is not found, the calculator still searches numerically. It evaluates f(x) across the interval. It then finds places where f(x) equals the requested y value. This makes the tool practical for classroom examples and quick checks.
Reading the Result
The result section appears above the form after submission. It shows the entered function, target y value, symbolic inverse when available, sample range, one-to-one trend, and inverse roots. The table displays sampled x and f(x) values. It also adds roots that match your target. Use the export buttons to save work for assignments, notes, or later review.
Best Entry Tips
Write multiplication clearly. Use 2*x instead of 2x. Use x^2 for powers. Use sqrt(x), ln(x), log(x), exp(x), sin(x), cos(x), and tan(x) for common functions. Pick a domain that covers the expected answer. A smaller interval often gives cleaner branch behavior. Always compare the inverse answer by substituting it back into the original function.
FAQs
What is an inverse function?
An inverse function reverses the original function. If f(x) gives y, then f⁻¹(y) gives the original x value.
Why do some functions have two inverse answers?
Some functions are not one-to-one on their full domain. A square function can produce the same y value from two different x values.
How do I enter powers?
Use the caret symbol. For example, enter x^2 for x squared and x^3 for x cubed.
Can I use logarithms?
Yes. Use ln(x) for natural logarithm. Use log(x) for base ten logarithm. Keep the logarithm input positive.
Why should I set a domain?
The domain controls the numeric search interval. It also helps separate branches when a function has more than one inverse value.
What does one-to-one mean?
One-to-one means every y value comes from only one x value. Such functions have clean inverse functions over that interval.
What if no exact inverse value appears?
The calculator reports the nearest sampled value. Try widening the domain, narrowing the branch, or checking the entered function.
Can I download my result?
Yes. Use the CSV button for spreadsheet work. Use the PDF button for a simple printable report.