Inverse Function Calculator

Calculator Inputs

Choose a function family, enter parameters, and compute the inverse formula, inverse value, domain, range, and graph reflection.

Function family

Target output y

Quadratic branch

a

b

c

d

Exponent n

Log or exponent base

Log shift

Quadratic h

Quadratic k

Graph of Function and Inverse

The inverse reflects across y = x

The chart plots the original function, its inverse, and the symmetry line y = x. Undefined points are skipped automatically.

Formula Used

Linear: f(x) = ax + b → f⁻¹(x) = (x - b) / a

Fractional linear: f(x) = (ax + b)/(cx + d) → f⁻¹(x) = (dx - b)/(a - cx)

Power: f(x) = ax^n + b → f⁻¹(x) = ((x - b)/a)^(1/n)

Exponential: f(x) = a·base^x + b → f⁻¹(x) = log_base((x - b)/a)

Logarithmic: f(x) = a·log_base(x - shift) + b → f⁻¹(x) = shift + base^((x - b)/a)

Restricted quadratic: f(x) = a(x - h)^2 + k → f⁻¹(x) = h ± √((x - k)/a)

How to Use This Calculator

Select the function family that matches your algebra problem.
Enter the coefficients and any model-specific parameters.
Type the output value y you want to invert.
Choose the branch when using the restricted quadratic model.
Press Calculate Inverse to generate results.
Review the inverse expression, inverse value, and forward-check result.
Inspect the graph to confirm reflection around y = x.
Use CSV or PDF export for records, classwork, or review.

Example Data Table

Function type	Input model	Target y	Inverse result	Notes
Linear	f(x) = 2x + 3	11	4	Swap x and y, then isolate.
Exponential	f(x) = 3·2^x + 1	25	3	Subtract one, divide by three, then log base two.
Logarithmic	f(x) = 2·log₁₀(x - 1) + 5	7	11	Exponentiate after isolating the logarithm.
Restricted quadratic	f(x) = (x - 1)^2 - 2, x ≥ 1	7	4	Use the positive branch to keep one-to-one behavior.

Frequently Asked Questions

1. What does an inverse function do?

An inverse function reverses the original mapping. If f transforms x into y, then f⁻¹ transforms that same y back into x, provided the function is one-to-one on its domain.

2. Why must some functions be restricted first?

Functions like quadratics fail the horizontal line test on their full domain. Restricting the domain to one branch makes the function one-to-one, which allows a valid inverse.

3. Why is the line y = x shown?

A function and its inverse are reflections across the line y = x. If the graph mirrors cleanly around that line, the inverse setup is behaving as expected.

4. What makes a fractional linear function invertible?

The condition ad - bc ≠ 0 is required. Without it, the transformation collapses information and cannot be reversed into a unique original input.

5. Can this solve every possible algebraic inverse?

No. This tool focuses on common invertible families used in coursework and analysis. It does not perform general symbolic inversion for arbitrary custom expressions.

6. Why does the calculator sometimes show “not defined”?

That appears when your chosen y lies outside the function’s range, when a logarithm receives a nonpositive argument, or when division by zero would occur.

7. What does the forward check mean?

The tool computes f(f⁻¹(y)). A correct inverse should return the original target y, aside from small rounding differences caused by numerical precision.

8. When should I use CSV or PDF export?

Use CSV for storing values in spreadsheets or reports. Use PDF when you need a printable summary for class notes, homework checks, or documentation.