Inverse function

Definition

An inverse function reverses the effect of the original function, denoted as $f^{-1}(x)$. For a function $f(x)$ to have an inverse, it must be bijective (both injective and surjective).

5 Must Know Facts For Your Next Test

1. To verify if two functions are inverses of each other, check if $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.
2. The graph of an inverse function is a reflection of the graph of the original function across the line $y=x$.
3. A function has an inverse if and only if it is one-to-one (bijective).
4. The domain of the original function becomes the range of its inverse, and vice versa.
5. Inverse functions undo each other's operations; for example, if $f(x) = 2x + 3$, then $f^{-1}(x) = \frac{x - 3}{2}$.

Review Questions

• What condition must a function meet to have an inverse?
• How do you algebraically verify that two functions are inverses?
• What is the graphical relationship between a function and its inverse?

Related terms

Bijective Function: A function that is both injective (one-to-one) and surjective (onto), ensuring every element in the codomain is mapped by exactly one element in the domain.

Injective Function: A type of function where each element in the domain maps to a unique element in the codomain. Also known as one-to-one.

Surjective Function: A type of function where every element in the codomain has at least one corresponding element in the domain. Also known as onto.