Inverse function

Definition

An inverse function is a function that reverses the effect of the original function. If $f(x)$ is a function, then its inverse $f^{-1}(x)$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

5 Must Know Facts For Your Next Test

1. An inverse function exists only if the original function is one-to-one (bijective).
2. The graph of an inverse function is the reflection of the graph of the original function across the line $y=x$.
3. To find an inverse function algebraically, solve the equation $y=f(x)$ for $x$ in terms of $y$, and then interchange $x$ and $y$.
4. Not all functions have inverses; a horizontal line test can determine if a function has an inverse.
5. The composition of a function and its inverse results in the identity function: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

Review Questions

• What conditions must be met for a function to have an inverse?
• How do you find the inverse of a given function algebraically?
• Describe how you can use graphs to identify if two functions are inverses.

Related terms

One-to-One Function:

A function where each output value corresponds to exactly one input value, ensuring it passes both vertical and horizontal line tests.

Horizontal Line Test:

A graphical test used to determine if a function has an inverse by checking if any horizontal line intersects the graph at most once.

$\text{Composite Function}$: $\text{A}$ $\text{function formed by applying one}$ $\text{function}$ $\text{to}$ $\text{the result}$ $\text{of another}$ $\text{function:}\\ $(g \circ f)(x) = g(f(x)).