Function Inverse Calculator With Steps

Calculator Form

Function type

Graph minimum x

Graph maximum x

Graph points

Verification input x

Linear model: f(x) = ax + b

a

b

Rational model: f(x) = (ax + b) / (cx + d)

a

b

c

d

Quadratic model: f(x) = a(x - h)² + k

a

h

k

Branch

Exponential model: f(x) = A * B^{(mx + n)} + k

A

B

m

n

k

Logarithmic model: f(x) = A * log_B(mx + n) + k

A

B

m

n

k

How to use this calculator

Choose the function family that matches your expression.
Enter the coefficients for that family.
Set the graph window and point count.
Enter an optional verification input.
Press Calculate inverse to show the inverse above the form.
Review the steps, restrictions, sample points, and mirrored graph.
Use CSV or PDF export when you need a saved report.

Formula used

Linear: Swap x and y in y = ax + b, then solve for y. The inverse becomes f^-1(x) = (x - b) / a.

Rational: Start from y = (ax + b) / (cx + d). Swap variables, cross multiply, group the y terms, factor, and isolate y.

Quadratic with branch: Use y = a(x - h)² + k. A branch restriction makes the function one-to-one, so the square root sign is chosen consistently.

Exponential: For y = A * B^{(mx + n)} + k, isolate the exponential term, then apply logarithms to solve for the variable.

Logarithmic: For y = A * log_B(mx + n) + k, isolate the logarithm, convert to exponential form, then solve for the variable.

Example data table

Model	Original function	Inverse function	Main restriction
Linear	`2x + 3`	`(x - 3) / 2`	`a != 0`
Rational	`(2x + 1) / (x - 4)`	`(4x + 1) / (x - 2)`	`ad - bc != 0`
Quadratic	`(x - 1)^2 + 2`	`1 + sqrt(x - 2)`	Use one branch only
Exponential	`2 * 3^x`	`log(x / 2) / log(3)`	`B > 0, B != 1`
Logarithmic	`2 * log₁₀(x + 1)`	`10^(x / 2) - 1`	`x > -1`

Frequently asked questions

1. Why do some functions need a restriction first?

An inverse exists only when each output comes from one input. Quadratic expressions usually fail that test, so a branch restriction makes them one-to-one.

2. What does the mirrored graph show?

It shows the original function, its inverse, and the line y = x. True inverse pairs reflect across that line.

3. Why does the rational model block some values?

A rational function can hit a zero denominator. That creates a forbidden input, and its inverse inherits a matching forbidden output.

4. Why is the exponential inverse logarithmic?

Exponentials and logarithms undo each other. After isolating the exponential term, a logarithm removes the exponent and reveals the variable.

5. Why is the logarithmic inverse exponential?

A logarithm states an exponent in another form. Converting that statement back to exponential notation isolates the original variable cleanly.

6. What does the verification box do?

It tests one input value. The tool computes f(x), then applies the inverse to that output and checks whether it returns the starting input.

7. Can I export the result after calculation?

Yes. Use the CSV button for tabular values and the PDF button for a compact text report with formulas, restrictions, and steps.

8. Does this handle every possible function?

No. It focuses on common invertible families used in classwork and practice. More complex symbolic inverses usually need a computer algebra system.