Inverse Function Calculator

How to Find the Inverse of a Function

This inverse function calculator automates the algebraic process, which follows the same three steps every time:

1. Write the function as y = f(x).
2. Swap x and y — replace every x with y and every y with x.
3. Solve for y — the result is f⁻¹(x).

This calculator automates that process for the four most common function families: linear, power, exponential, and logarithmic. Select a function type from the dropdown, enter the relevant parameters, and the calculator returns both the original function notation and the inverse, along with domain and range restrictions.

Inverse Function Formulas by Type

Each function family has its own inverse formula:

• Linear: f(x) = mx + b → f⁻¹(x) = (x − b) / m. Requires m ≠ 0.
• Power: f(x) = xⁿ → f⁻¹(x) = x^(1/n) = ⁿ√x. For even n, restrict the domain to x ≥ 0 to ensure the function is one-to-one.
• Exponential: f(x) = aˣ → f⁻¹(x) = log_a(x). The domain of the inverse is x > 0 because logarithms are not defined for non-positive values.
• Logarithmic: f(x) = log_a(x) → f⁻¹(x) = aˣ. These two are exact inverses of each other — this is the definition of the logarithm.

Domain and Range of Inverse Functions

The domain and range of an inverse function are the reverse of the original function. This symmetry is one of the most useful properties in algebra:

• Domain of f⁻¹ = Range of f
• Range of f⁻¹ = Domain of f

For example, f(x) = x² restricted to x ≥ 0 has domain [0, ∞) and range [0, ∞). Its inverse, f⁻¹(x) = √x, also has domain [0, ∞) and range [0, ∞). Without the domain restriction, x² is not one-to-one (both x = 2 and x = −2 give f(x) = 4), so no inverse exists over all real numbers.

For further exploration of function relationships, see our quadratic formula calculator or the slope calculator. When you need to solve for multiple unknowns at once, our systems of equations calculator handles 2×2 and 3×3 linear systems with step-by-step Gaussian elimination.

The Horizontal Line Test

A function has an inverse if and only if it passes the horizontal line test: no horizontal line can intersect the graph of f(x) at more than one point. This guarantees that every output value corresponds to exactly one input value (the function is one-to-one).

Functions that fail the test — such as f(x) = x² over all reals, or f(x) = sin(x) — can still be inverted if you restrict the domain to a portion where the function is one-to-one. For x², restricting to x ≥ 0 allows the inverse √x. For sin(x), restricting to [−π/2, π/2] gives the inverse arcsin(x).

Verifying an Inverse Function

You can verify that f⁻¹(x) is correct by composing the two functions: if f(f⁻¹(x)) = x and f⁻¹(f(x)) = x for all x in the domain, then f⁻¹ is the true inverse. For f(x) = 2x + 3 and f⁻¹(x) = (x − 3) / 2:

• f(f⁻¹(x)) = 2 · ((x − 3) / 2) + 3 = (x − 3) + 3 = x ✓
• f⁻¹(f(x)) = ((2x + 3) − 3) / 2 = 2x / 2 = x ✓

Both compositions return x, confirming the inverse is correct.

Real-World Applications of Inverse Functions

Inverse functions appear in virtually every quantitative field:

• Decibels and power: converting between dB and power ratio uses logarithms and their exponential inverses.
• Compound interest: solving for the interest rate or time period given a final amount requires inverting an exponential growth formula.
• Unit conversions: every unit conversion formula has an inverse — the formula to convert Celsius to Fahrenheit is the inverse of Fahrenheit to Celsius.
• Cryptography: public-key encryption relies on mathematical operations that are easy to perform in one direction but hard to invert without a key (trapdoor functions).