Range of a function

In mathematics, the range of a function may refer to either of two closely related concepts:

• the codomain of the function, or
• the image of the function.

In some cases the codomain and the image of a function are the same set; such a function is called surjective or onto. For any non-surjective function ${\displaystyle f:X\to Y,}$ the codomain ${\displaystyle Y}$ and the image ${\displaystyle {\tilde {Y}}}$ are different; however, a new function can be defined with the original function's image as its codomain, ${\displaystyle {\tilde {f}}:X\to {\tilde {Y}}}$ where ${\displaystyle {\tilde {f}}(x)=f(x).}$ This new function is surjective.