Question:
Let $A=\{x \in R:-1 \leq x \leq 1\}=B$ and $C=\{x \in R: x \geq 0\}$ and
let $S=\left\{(x, y) \in A \times B: x^{2}+y^{2}=1\right\}$ and $S_{0}=\left\{(x, y) \in A \times C: x^{2}+y^{2}=1\right\} .$ Then,
(a) $S$ defines a function from $A$ to $B$
(b) So defines a function from $A$ to $C$
(c) $S_{0}$ defines a function from $A$ to $B$
(d) $S$ defines a function from $A$ to $C$
Solution:
(a) S defines a function from A to B
Let $x \in A$
$\Rightarrow-1 \leq x \leq 1$
Now, $x^{2}+y^{2}=1$
$\Rightarrow y^{2}=1-x^{2}$
$\Rightarrow y=\pm \sqrt{1-x^{2}}$
$\Rightarrow-1 \leq y \leq 1$
$\therefore y \in B$
Thus, $S$ defines a function from $A$ to $B$.