Calculate the inverse of linear, quadratic, exponential, logarithmic, power, and rational functions, and evaluate the inverse at any value of x.
Inverse Function Formula
An inverse function reverses the action of another function, so that if y = f(x) then x = f⁻¹(y). To find an inverse algebraically you swap x and y in the equation and solve for y. The original function must be one-to-one for its inverse to also be a function. The exact form of the inverse depends on the type of function you start with.
Linear function:
f(x) = ax + b => f⁻¹(x) = (x − b) / a
Quadratic function in vertex form (with a restricted domain):
f(x) = a(x − h)² + k => f⁻¹(x) = h ± √((x − k) / a)
Exponential function:
f(x) = a·bˣ + c => f⁻¹(x) = log_b((x − c) / a)
Logarithmic function:
f(x) = a·log_b(x) + c => f⁻¹(x) = b^((x − c) / a)
Power function:
f(x) = a·xⁿ + c => f⁻¹(x) = ((x − c) / a)^(1/n)
Rational (linear fractional) function:
f(x) = (ax + b) / (cx + d) => f⁻¹(x) = (b − dx) / (cx − a)
In these formulas a, b, c, d, h, and k are constant coefficients, n is the exponent, x is the input value, f(x) is the original function, and f⁻¹(x) is its inverse. For quadratic and even-power functions you restrict the domain so the function is one-to-one. For exponential and logarithmic functions the base b must be positive and not equal to 1, and for the rational form the determinant ad − bc must not be 0.
Common Functions and Their Inverses
| Function f(x) | Inverse f⁻¹(x) | Condition |
|---|---|---|
| x + b | x − b | all real x |
| ax + b | (x − b) / a | a ≠ 0 |
| x² | √x | x ≥ 0 |
| xⁿ | x^(1/n) | n ≠ 0; choose an even-integer branch; noninteger powers use nonnegative x, excluding zero for negative n |
| eˣ | ln(x) | x > 0 |
| bˣ | log_b(x) | b > 0, b ≠ 1, x > 0 |
| ln(x) | eˣ | all real x |
| 1 / x | 1 / x | x ≠ 0 |
The domain of f becomes the range of f⁻¹, and the range of f becomes the domain of f⁻¹. The graph of an inverse is the reflection of the original function across the line y = x.
Example Problems
Example 1 (linear). Find the inverse of f(x) = 3x − 6. Swap x and y to get x = 3y − 6, then solve for y: y = (x + 6) / 3. So f⁻¹(x) = (x + 6) / 3. To check this with the calculator, choose Linear, set a = 3 and b = −6, and evaluate at x = 3 to confirm f⁻¹(3) = 3.
Example 2 (rational). Find the inverse of f(x) = (2x + 1) / (x − 3). Here a = 2, b = 1, c = 1, and d = −3, so f⁻¹(x) = (b − dx) / (cx − a) = (1 + 3x) / (x − 2), which is the same as (3x + 1) / (x − 2). Choose Rational and enter these values to see the result.
Frequently Asked Questions
What is an inverse function? An inverse function undoes the original function. If you apply f and then apply f⁻¹, you return to your starting value, so f⁻¹(f(x)) = x for every x in the domain.
Does every function have an inverse? No. Only one-to-one functions have an inverse that is itself a function. A function like x² is not one-to-one over all real numbers, so you restrict the domain (for example to x ≥ 0) before taking the inverse.
How do you check that an inverse is correct? Compose the two functions. If f(f⁻¹(x)) = x and f⁻¹(f(x)) = x, then the inverse is correct. You can also evaluate a point with this calculator and confirm the original function maps it back.
