# Image of Element under Mapping/Examples

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## Examples of Image of Element under Mapping

### Image of $2$ under $x \mapsto x^4 - 1$

Let $f: \R \to \R$ be the mapping defined as:

$\forall x \in \R: \map f x = x^4 - 1$

The image of $2$ is:

$\map f 2 = 15$

### Images of Various Numbers under $x \mapsto x^2 + 2 x + 1$ in $\closedint 0 1$

Let $f: \closedint 0 1 \to \R$ be the mapping defined as:

$\forall x \in \closedint 0 1: \map f x = x^2 + 2 x + 1$

where $\closedint 0 1$ denotes the closed real interval from $0$ to $1$.

The images of various real numbers under $f$ are:

 $\displaystyle \map f 0$ $=$ $\displaystyle 0^2 + 2 \times 0 + 1$ $\displaystyle = 1$ $\displaystyle \map f 1$ $=$ $\displaystyle 1^2 + 2 \times 1 + 1$ $\displaystyle = 4$ $\displaystyle \map f {\dfrac 1 2}$ $=$ $\displaystyle \paren {\dfrac 1 2}^2 + 2 \times \dfrac 1 2 + 1$ $\displaystyle = 2 \tfrac 1 4$ $\displaystyle \map f 2$  $\displaystyle \text {is undefined}$ $\displaystyle \text {as 2 is not in the domain of f}$ $\displaystyle \map f {-1}$  $\displaystyle \text {is undefined}$ $\displaystyle \text {as -1 is not in the domain of f}$