Let A -1-1- B -1-1- C -0- - Define R 1- x - y - A - B - x 2- y 2-1 and R 2- x - y - A - 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-

The correct option is B R_2 is a function and range is [0,1]Consider R_1 For nbsp;x=0, y can have two values, either 1 or 1. This violates the definition of ...

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Byju's Answer

Standard XII

Mathematics

Range

Let A =[-1,1]...

Question

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

R_{1}

is a function and range is

[0, 1]

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R_{2}

is a function and range is

[0, 1]

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R_{1}

is a function and range is

[- 1, 1]

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R_{2}

is a function and range is

[- 1, 1]

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Solution

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

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

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Similar questions

Q. Let

A = [- 1, 1], B = [- 1, 1], C = [0, \infty)

. Define

R_{1} = {(x, y) \in A \times B : x^{2} + y^{2} = 1}

and

R_{2} = {(x, y) \in A \times C : x^{2} + y^{2} = 1}

. Then choose the correct option:

Q. Let A = [-1, 1], B = [-1, 1], C =

[0, \infty)

. Let

{(x, y) ϵ A \times B : x^{2} + y^{2} = 1}

and

R_{2} = {(x, y) ϵ A \times C : x^{2} + y^{2} = 1}

Q. Range of the function

y = \frac{2^{x} - 2^{- x}}{2^{x} + 2^{- x}}

$R a n g e o f t h e f u n c t i o n y = \frac{2^{x} - 2^{- x}}{2^{x} + 2^{- x}} i s$

Q. Range of the function

y = \frac{2^{x} - 2^{- x}}{2^{x} + 2^{- x}}

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Let A=[−1,1], B=[−1,1], C=[0,∞). Define R1={(x,y)∈A×B:x2+y2=1} and R2={(x,y)∈A×C:x2+y2=1}. Then choose the correct option:

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