Let

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 $B$.

 

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