Question

From the given functions, which of the following is the point function

• A. $\sqrt{x-2}+\sqrt{-x+2}+8$
• B. $\ln (1-x)+\sqrt{x-1}$
• C. $\sqrt{x+2}+{\sin }^{-1}x$
• D. ${\sin }^{-1}x+{\text{cosec}}^{-1}x$

Answer 1

The correct option is A $\sqrt{x-2}+\sqrt{-x+2}+8$

We know, function having a single point in its domian is a point function.

Let $f_1\left(x\right)=\sqrt{x-2}+\sqrt{-x+2}+8$
For domain, $x-2\ge 0$ and $-x+2\ge 0$
$\Rightarrow x\ge 2$ and $x\le 2$
Domain of $f_1\in \left\{2\right\}$

Let $f_2\left(x\right)=\ln (1-x)+\sqrt{x-1}$
For domain, $1-x>0$ and $x-1\ge 0$
$\Rightarrow x<1$ and $x\ge 1$
Domain of $f_2\in \left\{\varphi \right\}$

Let $f_3\left(x\right)=\sqrt{x+2}+{\sin }^{-1}x$
For domain, $x+2\ge 0$ and $x\in [-1,1]$
$\Rightarrow x\ge -2$ and $x\in [-1,1]$
Domian of $f_3\in [-1,1]$

Let $f_4\left(x\right)={\sin }^{-1}x+{\text{cosec}}^{-1}x$
For domain, $x\in [-1,1]$ and $x\in R-(-1,1)$
Domain of $f_4\in \{-1,1\}$