Inverse function

Definition

An inverse function reverses the operation of a given function. If $f(x)$ is a function, 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. The inverse of a function $f(x)$ is denoted as $f^{-1}(x)$.
2. To find an inverse function, swap the roles of $x$ and $y$ in the equation and solve for y.
3. A function must be one-to-one (bijective) to have an inverse.
4. Graphically, the graph of an inverse function is the reflection of the original function across the line $y = x$.
5. If a function passes the horizontal line test, it has an inverse.

Review Questions

• What is the notation for an inverse function?
• Describe how you would find the inverse of a given function algebraically.
• How can you use graphs to determine if a function has an inverse?

Related terms

One-to-One Function: A one-to-one function is one where each input maps to exactly one unique output.

Horizontal Line Test: A test used to determine if a function has an inverse by checking if any horizontal line intersects the graph more than once.

Bijective Function: A bijective function is both injective (one-to-one) and surjective (onto), meaning it maps every element of its domain to a unique element in its codomain.