What is an Inverse Function?
If f(x) maps every input x to an output y, then the inverse function f⁻¹(x) reverses that mapping — it takes y back to x. Formally: if f(a) = b, then f⁻¹(b) = a.
Calculate Inverse Function
Enter your function below — the calculator will find its inverse f⁻¹(x).
Supports: linear (ax+b), quadratic (ax²+bx+c), √x, x^n, eˣ, ln(x), log(x)
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Your inverse function will appear here
How to Find an Inverse
Replace f(x) with y, swap x and y, then solve for y. The result is f⁻¹(x). A function must be one-to-one (bijective) for a true inverse to exist.
Graphical Interpretation
The graph of f⁻¹(x) is the reflection of f(x) across the line y = x. This means every point (a, b) on f(x) corresponds to (b, a) on f⁻¹(x).
Common Inverse Functions Reference
| Function f(x) | Inverse f⁻¹(x) | Domain of f⁻¹ | Type |
|---|---|---|---|
| ax + b | (x − b) / a | ℝ (all reals) | Linear |
| x² | √x | x ≥ 0 | Quadratic |
| x³ | ∛x | ℝ | Cubic |
| eˣ | ln(x) | x > 0 | Exponential |
| ln(x) | eˣ | ℝ | Logarithmic |
| 10ˣ | log₁₀(x) | x > 0 | Exponential (base 10) |
| sin(x) | arcsin(x) | −1 ≤ x ≤ 1 | Trigonometric |
| cos(x) | arccos(x) | −1 ≤ x ≤ 1 | Trigonometric |
| tan(x) | arctan(x) | ℝ | Trigonometric |
| 1/x | 1/x | x ≠ 0 | Reciprocal |
Learn More About Inverse Functions
Dive deep into inverse function theory, properties, worked examples, and real-world applications in our blog.