Question

Find the domain and range of the function $f\left(x\right)=\frac{1}{\sqrt{x-5}}$

Answer 1

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

Clearly f(x) is defined if x-5 > 0

$\Rightarrow x>5$

$\therefore Domainoffis(5,\infty )$

Since, f(x) is defined for x > 5

$x-5>0$

$\Rightarrow \sqrt{x-5}>0$

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

$(ifa>0then\frac{1}{a}>0)$

$\Rightarrow f\left(x\right)>0$

$\therefore Rangeoffis(0,\infty )$

Hence, domain of $f\left(x\right)is(5,\infty )$
and Range of $f\left(x\right)is(0,\infty )$