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

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 $f$}$

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 $2.5$