Image of Element under Mapping/Examples

Examples of Image of Element under Mapping

Arbitrary Example

Let:

\(\ds A\)

\(=\)

\(\ds \set {a, b, c, d}\)

\(\ds B\)

\(=\)

\(\ds \set {w, x, y, z}\)

Let $f: A \to B$ be a mapping defined such that:

$\map f a = z$

Then $z$ is the image of $a$ under $f$.

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:

\(\ds \map f 0\)

\(=\)

\(\ds 0^2 + 2 \times 0 + 1\)

\(\ds = 1\)

\(\ds \map f 1\)

\(=\)

\(\ds 1^2 + 2 \times 1 + 1\)

\(\ds = 4\)

\(\ds \map f {\dfrac 1 2}\)

\(=\)

\(\ds \paren {\dfrac 1 2}^2 + 2 \times \dfrac 1 2 + 1\)

\(\ds = 2 \tfrac 1 4\)

\(\ds \map f 2\)

\(\)

\(\ds \text {is undefined}\)

\(\ds \text {as $2$ is not in the domain of $f$}\)

\(\ds \map f {-1}\)

\(\)

\(\ds \text {is undefined}\)

\(\ds \text {as $-1$ is not in the domain of $f$}\)