# 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:

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

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

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

Let:

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

Then:

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

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:

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

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

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

### Image of $\closedint 1 2$ under $\map f x = x^2 - x - 2$

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

$\forall x \in \R: \map f x = x^2 - x - 2$

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

$f \closedint 1 2 = \closedint {-2} 0$

### Image of $\openint {-1} 1$ under $\map f x = x^2 - x - 2$

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

$\forall x \in \R: \map f x = x^2 - x - 2$

The image of the open interval $\openint {-1} 1$ is:

$f \sqbrk {\openint {-1} 1} = \hointr {-\dfrac 9 4} 0$