Calculator
Graph
The chart plots sample points from the selected function.
Example Data Table
| Function model | Original function | Inverse form | Target y | Returned x |
|---|---|---|---|---|
| Linear | f(x) = 2x + 5 | f⁻¹(x) = (x - 5) / 2 | 15 | 5 |
| Fraction | f(x) = (3x + 6) / 2 | f⁻¹(x) = (2x - 6) / 3 | 12 | 6 |
| Reciprocal | f(x) = 8 / (x - 1) + 4 | f⁻¹(x) = 1 + 8 / (x - 4) | 8 | 3 |
| Quadratic branch | f(x) = 2(x - 1)² + 4 | f⁻¹(x) = 1 + √((x - 4) / 2) | 12 | 3 |
Formula Used
Linear: For f(x) = ax + b, the inverse is f⁻¹(x) = (x - b) / a.
Fraction: For f(x) = (ax + b) / c, the inverse is f⁻¹(x) = (cx - b) / a.
Reciprocal: For f(x) = a / (x - h) + k, the inverse is f⁻¹(x) = h + a / (x - k).
Restricted quadratic: For f(x) = a(x - h)² + k, the inverse branch is f⁻¹(x) = h ± √((x - k) / a).
The plus or minus sign depends on the selected branch. A quadratic must be restricted before it has a valid inverse function.
How to Use This Calculator
- Select the function model that matches your equation.
- Enter the required coefficients and shift values.
- Enter a target output value for y.
- Choose the correct branch for quadratic equations.
- Press the calculate button.
- Review the inverse rule, domain, range, graph, and algebra steps.
- Use CSV or PDF export for records and assignments.
Understanding Multi Step Function Inverses
What an Inverse Function 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 calculator is built for multi step expressions. It supports common classroom forms. It also checks useful restrictions.
Why Steps Matter
Inverse work is not only about the final answer. Each algebra move matters. You usually start by replacing f(x) with y. Then you solve for x. After that, you swap x and y. The final expression becomes the inverse rule. Showing the steps helps you find sign mistakes.
Domain and Range
Domain and range are very important in inverse questions. A linear function often has all real numbers for both. A reciprocal function excludes values that create division by zero. A quadratic needs a branch restriction. Without this restriction, one output can come from two inputs.
Verification
A good inverse should pass a reverse check. Place the calculated x into the original function. The answer should match the target y. You can also compose the functions. For true inverses, f(f⁻¹(x)) returns x where both expressions are defined.
Graph Support
The graph gives a visual check. Inverse functions reflect across the line y = x. This calculator plots sample points from the chosen original function. The table and graph together make patterns easier to understand. They also help explain restrictions.
FAQs
1. What is a function inverse?
A function inverse reverses the original rule. It changes an output back into the original input when the function is one-to-one.
2. Why do I swap x and y?
You swap x and y because the inverse changes the direction of the relationship. Inputs become outputs, and outputs become inputs.
3. Can every function have an inverse?
No. A function must be one-to-one. If one output comes from more than one input, it needs restriction or has no inverse function.
4. Why does a quadratic need a branch?
A full quadratic usually fails the horizontal line test. A left or right branch makes it one-to-one, so an inverse can be written.
5. What does target y mean?
Target y is the output value you want to reverse. The calculator finds the input x that would produce that output.
6. What happens if division by zero occurs?
The calculator marks the result invalid. Division by zero is undefined, so that value must be excluded from the domain or range.
7. How do I check my inverse?
Substitute the inverse result into the original function. If it returns the target output, the reverse calculation is consistent.
8. Can I export the result?
Yes. Use the CSV button for spreadsheet records. Use the PDF button for printable notes, homework, reports, or study files.