# 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:

\(\ds \map f 0\) | \(=\) | \(\ds 0^2 + 2 \times 0 + 1\) | \(\ds = 1\) | |||||||||||

\(\ds \map f 1\) | \(=\) | \(\ds 1^2 + 2 \times 1 + 1\) | \(\ds = 4\) | |||||||||||

\(\ds \map f {\dfrac 1 2}\) | \(=\) | \(\ds \paren {\dfrac 1 2}^2 + 2 \times \dfrac 1 2 + 1\) | \(\ds = 2 \tfrac 1 4\) | |||||||||||

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

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

## Sources

- 1963: Morris Tenenbaum and Harry Pollard:
*Ordinary Differential Equations*... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $2 \text B$: The Meaning of the Term*Function of One Independent Variable*: Example $2.5$