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Moderate

Image of a Map

ABSALG-NTWQGX

Let $S$ and $T$ be two (non-empty) sets, and let $f:S\rightarrow T$ be a function with domain $S$. The set $T$ is called the codomain of $f$.

The image of $f$, or, synonymously, the image of $S$ (when $f$ is understood) is the set $f(S)$ defined by:

$$f(S)=\{t\in T\mid f(s)=t,\;{\rm for\;some}\;s\in S\}$$

If $A\subseteq S$ then the image of $A$ is defined by:

$$f(A)=\{t\in T\mid f(s)=t,\;{\rm for\;some}\;s\in A\}$$

The image $f(S)$ of $S$ is sometimes called the range of $f$.

Which of the following is true? For each choice, by image we mean $f(S)$.

(Notation: $i=\sqrt{-1}$, ${\mathbb R}$ real numbers, ${\mathbb C}$ complex numbers.)

A

$f:S=\{1,-i\}\rightarrow T=\{t\in{\mathbb C}\mid |t|=1\}$, given by:

$f(s)=s^3$

...has image $\{1, -i\}$.

B

$f:S=\{1,i, -1, -i\}\rightarrow \{1, i, -1, -i\}$, given by:

$f(s)=s^{1/3}$

...has no well-defined image as it is not a function.

C

$f:S=\{\theta\in{\mathbb R}\mid -\pi<\theta\le\pi\}\rightarrow T^2=\{(t,w)\in{\mathbb C}^2\mid |t|=|w|=1\}$, given by:

$f(\theta)=(\cos\theta+i\sin\theta,\cos\theta-i\sin\theta)$

...has image $T^2$.

D

$f:S=\{\theta\in{\mathbb R}\mid -\pi<\theta\le\pi\}\rightarrow H=\{(t,w)\in{\mathbb C}^2\mid tw=1\}$, given by:

$f(\theta)=(\cos\theta+i\sin\theta,\cos\theta-i\sin\theta)$

...has image $H$.

E

$f:S=\{\theta\in{\mathbb R}\mid -\pi<\theta\le\pi\}\rightarrow T^2=\{(t,w)\in{\mathbb C}^2\mid |t|=|w|=1\}$, given by:

$f(\theta)=(\cos\theta+i\sin\theta,\cos\theta-i\sin\theta)$

...has image:

$\{(t,w)\in{\mathbb C}^2\mid w=t^{-1}\;{\rm and}\;|t|=1\}$.​