Question

Let $A=[-1,1],B=[-1,1],C=[0,\infty )$. Define $R_1=\left\{\right(x,y)\in A\times B:x^2+y^2=1\}$ and $R_2=\left\{\right(x,y)\in A\times C:x^2+y^2=1\}$. Then choose the correct option:

• A. $R_1$ is a function and range is $[0,1]$
• B. $R_2$ is a function and range is $[0,1]$
• C. $R_1$ is a function and range is $[-1,1]$
• D. $R_2$ is a function and range is $[-1,1]$

Answer 1

The correct option is B $R_2$ is a function and range is $[0,1]$

Consider $R_1$
For $x=0$, $y$ can have two values, either $1$ or $-1$.
This violates the definition of a function.
Hence, $R_1$ is not a function.

Consider $R_2$
$x^2+y^2=1$
$\Rightarrow y=\sqrt{1-x^2}[\because y\in C]$
For each $x\in [-1,1]$, only one value of $y$ is possible.
Hence, $R_2$ is a function.

Now, for the range of $R_2$
$-1\le x\le 1$
$\Rightarrow 0\le x^2\le 1$
$\Rightarrow 0\le 1-x^2\le 1$
$\Rightarrow 0\le \sqrt{1-x^2}\le 1$
$\therefore y\in [0,1]$
So, the range of $R_2$ is $[0,1]$