Question

Let $R=\left\{(x,y):x,y∈R,x^2+y^2≤25\right\}$ $R^{′}=\left\{(x,y):x,y∈R,yâ‰\yen \frac{4}{9}{x}^2\right\}\text{ then }$

• A. $\mathrm{dom}Râˆ\copyright R^{′}=[−4,4]$
• B. range $Râˆ\copyright R^{′}=[0,4]$
• C. range $Râˆ\copyright R^{′}=[0,5]$
• D. $Râˆ\copyright R^{′}$ defines a function.

Answer 1

Answer: (C)

The equation $x^2+y^2=25$ represents a circle with centre $(0,0)$ and radius 5 and the equation $y=\frac{4}{9}{x}^2$ represents a parabola with vertex $(0,0)$ and focus $(0,1/9)$.
Hence $Râˆ\copyright R^{′}$ is the set of points indicated in the Fig.1.27
$=\{(x,y):−3≤x≤3,0≤y≤3]\}$.
Thus dom $Râˆ\copyright R^{′}=[−3,3]$ and range $Râˆ\copyright R^{′}=[0,5]⊃[0,4]$
Since $(0,0)∈Râˆ\copyright R^{′}$ and $(0,5)∈Râˆ\copyright R^{′}$
$∴0$ is related to 0 as well as 5 .
Hence $Râˆ\copyright R^{′}$ doesn't define a function.