Image of Element under Mapping/Examples

From ProofWiki
Jump to navigation Jump to search

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:

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