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 $0$.
$\therefore R_{f} = [0, \infty)$