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:

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