1. What is a Function Inverse?

The inverse of a function f, denoted as f^{-1}, is a function that reverses the mapping of f. For a function f that maps x to y, the inverse function maps y back to x.

2. How Does the Calculator Work?

The calculator computes the inverse of a given function:

Where:

• \( f(x) \) — Original function
• \( f^{-1}(y) \) — Inverse function

Explanation: The calculator solves the equation f(x) = y for x in terms of y to find the inverse function.

3. Importance of Function Inverses

Details: Inverse functions are crucial in mathematics for solving equations, understanding function behavior, and applications in various fields including physics and engineering.

4. Using the Calculator

Tips: Enter the function in terms of x. Ensure the function is one-to-one to have a valid inverse.

5. Frequently Asked Questions (FAQ)

Q1: What functions have inverses?
A: Only one-to-one functions have inverses. A function is one-to-one if it passes the horizontal line test.

Q2: How do you find the inverse of a function?
A: Replace f(x) with y, swap x and y, then solve for y.

Q3: Can all functions be inverted?
A: No, only bijective functions (both injective and surjective) have inverses.

Q4: What is the domain of an inverse function?
A: The domain of f^{-1} is the range of f, and the range of f^{-1} is the domain of f.

Q5: Are there functions that are their own inverse?
A: Yes, for example f(x) = x and f(x) = 1/x (for x ≠ 0) are their own inverses.