Question

The domain and range of real function $f$ defined by $f\left(x\right)=\sqrt{x-1}$ is given by

• A. Domain $=(1,\infty )$, Range $=(0,\infty )$
• B. Domain $=[1,\infty )$, Range $=(0,\infty )$
• C. Domain $=[1,\infty )$, Range $=[0,\infty )$
• D. None of these

Answer 1

Answer: (C)

Given, $f\left(x\right)=\sqrt{x-1}$
$f\left(x\right)$ is defined when $x-1\ge 0$
$\Rightarrow x\ge 1$
$\therefore D_f=[1,\infty )$
For $x\ge 1$, we have $x-1\ge 0$
$\Rightarrow \sqrt{x-1}\ge 0$
$\Rightarrow f\left(x\right)\ge 0$
$\therefore f\left(x\right)$ can take all real values greater than or equal to $0$.
$\therefore R_f=[0,\infty )$