In mathematics, the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists if and only if f is bijective , and if it exists, is denoted by f − 1 . {\displaystyle f^{-1}.}

So the inverse of: 2x+3 is: (y−3)/2. The inverse is usually shown by putting a little "-1" after the function name, like this: f-1(y) We say "f inverse of y" So, the inverse of f (x) = 2x+3 is written: f-1(y) = (y-3)/2. (I also used y instead of x to show that we are using a different value.)

Nov 16, 2020 · An inverse of a mathematical function reverses the roles of y and x in the original function. Not all inverses of functions are true functions.

The inverse of a function \(f\) is another function \(f_{inv}\) defined so that \(f(f_{inv}(x)) = x\) and \(f_{inv}(f(x)) = x\) both hold. In words, the inverse function to \(f\) acting on \(f\) produces the identity function, \(x\).

Because the inverse of h ‍ is not a function, we say that h ‍ is non-invertible. In general, a function is invertible only if each input has a unique output. That is, each output is paired with exactly one input.

5 days ago · Given a function f (x) f (x), the inverse is written f^ {-1} (x) f −1(x), but this should not be read as a negative exponent. Generally speaking, the inverse of a function is not the same as its reciprocal.

May 1, 2025 · In this lesson, you will explore inverses of a given function that sometimes results in a non-function. You'll develop pairs of functions that are built around inverse operations, and explore their tables, graphs, and equations.

May 26, 2025 · This section explores inverse functions, explaining how to determine if a function has an inverse and how to find it. ... Figure \( \PageIndex{ 8 } \): Square and square-root functions on the non-negative domain. This relationship will be observed for all one-to-one functions, because it is a result of the function and its inverse swapping ...

Since a function must be well-defined, a function has an inverse function if and only if it is a bijective function. If a function is not bijective, it is common to restrict the domain and/or codomain to make it bijective to find an inverse function. Common examples:

Suppose and are both inverses of a function . By Theorem 5.1, the domain of is equal to the domain of , because both are the range of . This means the identity function applies both to the domain of and the domain of Thus. as required. We summarize the important properties of invertible functions in the following theorem.