# Image of Subset under Mapping/Examples

## Examples of Images of Subsets under Mappings

### Aribtrary Mapping from $\set {0, 1, 2, 3, 4, 5}$ to $\set {0, 1, 2, 3}$

Let:

 $\displaystyle S$ $=$ $\displaystyle \set {0, 1, 2, 3, 4, 5}$ $\displaystyle T$ $=$ $\displaystyle \set {0, 1, 2, 3}$

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

 $\displaystyle f \paren 0$ $=$ $\displaystyle 0$ $\displaystyle f \paren 1$ $=$ $\displaystyle 0$ $\displaystyle f \paren 2$ $=$ $\displaystyle 0$ $\displaystyle f \paren 3$ $=$ $\displaystyle 1$ $\displaystyle f \paren 4$ $=$ $\displaystyle 1$ $\displaystyle f \paren 5$ $=$ $\displaystyle 3$

Let:

 $\displaystyle A$ $=$ $\displaystyle \set {0, 3}$ $\displaystyle B$ $=$ $\displaystyle \set {0, 1, 3}$ $\displaystyle C$ $=$ $\displaystyle \set {0, 1, 2}$

Then:

 $\displaystyle f \sqbrk A$ $=$ $\displaystyle \set {0, 1}$ $\displaystyle f \sqbrk B$ $=$ $\displaystyle \set {0, 1}$ $\displaystyle f \sqbrk C$ $=$ $\displaystyle \set 0$

and:

$\Img f = \set {0, 1, 3}$

### Image of $\closedint {-3} 2$ under $x \mapsto x^4 - 1$

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

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

The image of the closed interval $\closedint {-3} 2$ is:

$f \sqbrk {\paren {\closedint {-3} 2} } = \closedint {-1} {80}$