Inverse Function Calculator
Instantly find the inverse of any mathematical function. Get step-by-step solutions for linear, quadratic, exponential, logarithmic, and rational functions with verification.
Function Input
Enter your function to find its inverse
Use standard mathematical notation: * for multiplication, ^ for powers, / for division
Restrict domain to ensure function is one-to-one
Inverse Function Result
f⁻¹(x) & verification steps
Enter a function like 2*x + 3 or x^2, then click Find Inverse Function to see the step-by-step solution and final inverse.
Standard Inverse Pairs
Common function-inverse pairs used in algebra, precalculus, and calculus courses.
| Original Function f(x) | Inverse Function f⁻¹(x) | Domain Restriction |
|---|---|---|
| mx + b (m ≠ 0) | (x – b)/m | None required |
| x² | √x | x ≥ 0 |
| x³ | ∛x | None required |
| eˣ | ln(x) | x > 0 |
| ln(x) | eˣ | x > 0 |
| aˣ (a > 0) | logₐ(x) | x > 0 |
| sin(x) | arcsin(x) | -π/2 ≤ x ≤ π/2 |
| cos(x) | arccos(x) | 0 ≤ x ≤ π |
| tan(x) | arctan(x) | -π/2 < x < π/2 |
| 1/x | 1/x | x ≠ 0 |
| √x | x² | x ≥ 0 |
| |x| | Not invertible | Requires piecewise definition |
Inverse Function FAQ
Learn about inverse functions, their properties, and how to work with them in mathematics.
An inverse function reverses the effect of the original function. If f(x) = y, then the inverse function f⁻¹(y) = x. For a function to have an inverse, it must be one-to-one (bijective), meaning each output corresponds to exactly one input. This ensures that the inverse is also a function.
To find the inverse of a function: 1) Replace f(x) with y, 2) Swap x and y variables, 3) Solve the equation for y, 4) Replace y with f⁻¹(x). The resulting function is the inverse, provided the original function is one-to-one. For example, if f(x) = 2x + 3, then y = 2x + 3 → x = 2y + 3 → y = (x – 3)/2 → f⁻¹(x) = (x – 3)/2.
No, only one-to-one functions (bijective functions) have inverses. A function is one-to-one if it passes the horizontal line test—no horizontal line intersects the graph more than once. Functions like f(x) = x² are not one-to-one over their entire domain but can have inverses when restricted to specific intervals (like x ≥ 0 for the square root function).
The graph of an inverse function is the reflection of the original function’s graph across the line y = x. This means if the point (a,b) is on the graph of f(x), then the point (b,a) will be on the graph of f⁻¹(x). The line y = x acts as the mirror line for this reflection, which visually demonstrates how inputs and outputs are swapped.
Two functions f and g are inverses if both f(g(x)) = x and g(f(x)) = x for all x in their respective domains. This composition test confirms that applying one function followed by the other returns the original input. For example, if f(x) = 2x + 3 and g(x) = (x – 3)/2, then f(g(x)) = 2((x-3)/2) + 3 = x – 3 + 3 = x, and g(f(x)) = (2x + 3 – 3)/2 = 2x/2 = x.
Many common functions like quadratics (f(x) = x²) or trigonometric functions are not one-to-one over their natural domains because they produce the same output for multiple inputs. By restricting the domain to an interval where the function is strictly increasing or decreasing, we ensure it becomes one-to-one and thus invertible. For example, sin(x) is restricted to [-π/2, π/2] to define arcsin(x).