Image of Element under Mapping/Examples/Images of Various Numbers under x^2+2x+1 in Limited Range

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Examples of Images of Elements under Mapping

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:

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


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