Question

Find the range of the following function :

$f\left(x\right)=\frac{1}{\sqrt{x-5}}$

• A. $(0,1)$
• B. $(-\infty ,\infty )$
• C. $(0,\infty )$
• D. $(-1,1)$
• E. $(-\infty ,0)$
• F. None of these

Answer 1

The correct option is C $(0,\infty )$

$f\left(x\right)=\frac{1}{\sqrt{x-5}}$

Clearly, $f\left(x\right)$ takes real values for all x satisfying $x-5>0⟹x>5⟹x\in (5,\infty )$

Therefore, $Domain\left(f\right)=(5,\infty )$

For any $x>5$, we have

$x-5>0⟹\sqrt{x-5}>0⟹\frac{1}{\sqrt{x-5}}>0$

$⟹f\left(x\right)>0$

Thus, $f\left(x\right)$ takes all real values greater than zero. Hence, Range (f) = $(0,\infty )$