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# Basic definitions

A function $f:X\rightarrow Y$ maps each input element $x$ of a given set $X$ to exactly one value $y$ of a set $Y$.

An element of $X$ cannot be mapped to more than one element of $Y$, but two elements of $X$ may be mapped to the same element of $Y$.

$f(x) = \vert x\vert$ associates both $1$ and $-1$to $1$. This is a function. The mapping that sends $1$ to both $-1$ and $1$ is not a function.

The value $f(x)$ is the image of $x$. The image of $x$ is often written as $y$. This image is expressed by $$f:x\mapsto f(x).$$

The set for which the function is defined is the domain. It is usually denoted by $D_f$. Often the function is written as $f:X\rightarrow Y$ where the set $X$ contains the domain. This is usually done for convenience.

The domain of the function $f(x) = 1/x$ is $D_f =\R\backslash\{0\}$.

The collection of values $f(x)$ for all the elements $x\in D_f$ is the range of $f$. This is contained in the set $Y$, which is called the target set or co-domain.

$f(x)= 1/(x-1)^2$ is defined for $x\ne 1$. Its range is $(0,+\infty)$.

The function $x\mapsto 1/(x-1)^2.$
The function is defined only on its domain. The resulting values give the range. The co-domain could be all of the $y$-axis.