First slide

Functions (XII)

Question

Find the range of the function $f (x) = x + \sqrt{x^{2}}$

Easy

A

$[0, \infty)$
B

$R$
C

$(0, \infty)$
D

None of these

Solution

$\begin{aligned} f (x) = x + \sqrt{x^{2}} = x + | x | \\ = \{\begin{cases} x + x, & x \geq 0 \\ x - x, & x < 0 \end{cases} \end{aligned}$
$\begin{array}{l} = \{\begin{matrix} 2 x, & x \geq 0 \\ 0, & x < 0 \end{matrix} \\ \Rightarrow f (x) takes values [0, \infty) \end{array}$

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