Function Value and Inverse Function Calculator

What does this tool do?

You type the definition of a function and the tool does three jobs step by step. The value tab finds f(a): it replaces every x with a and works out the result. The composition tab builds f(g(x)) and g(f(x)) from two functions and simplifies them. The inverse tab solves y = f(x) for x and swaps the letters x and y to give f⁻¹(x).

How to type

Write the function as f(x) = 2x^2 - 3x + 1 or simply as 2x^2 - 3x + 1. Use ^ for powers. The multiplication sign can be omitted. A value can be a whole number, a decimal, a fraction (1/2) or an expression such as pi/2 or sqrt(2). Separate several values with semicolons to evaluate them together.

Example: function value

For f(x) = 2x² − 3x + 1 find f(2). Replace x with 2: 2·2² − 3·2 + 1 = 8 − 6 + 1 = 3. When the value is a whole number or a fraction the result is given exactly as a fraction.

Composite functions

To find f(g(x)), every x in the definition of f is replaced with the expression g(x). For f(x) = 2x + 1 and g(x) = x² this gives f(g(x)) = 2x² + 1, while g(f(x)) = (2x + 1)² = 4x² + 4x + 1. The two results are usually different: the order matters.

Inverse function

For linear functions (ax + b) and for a linear expression over a linear one ((ax + b)/(cx + d)) the inverse is found in symbols. Example: in y = (2x + 1)/(x − 3) multiply both sides by the denominator, collect the terms with x and get x = (3y + 1)/(y − 2). Swapping the letters gives f⁻¹(x) = (3x + 1)/(x − 2). The tool checks the result with f(f⁻¹(x)) = x.

Limits

For other functions (such as x², x³ + x or eˣ) no symbolic inverse is given. Instead the equation f(x) = y is solved for the value of y you enter and the solutions are listed. If there are several, the function is not one to one and an inverse exists only on a restricted domain. For functions that are not polynomials, solutions are searched only between −10 and 10. Trigonometric functions take radians. Compositions with roots or trigonometry get basic simplification only.