Question

The domain of the function $f(x) = \frac{1}{\sqrt{x + |x|}}$ is

• A. $(-\infty, 0)$
• B. $(2, 5)$
• C. $(0, \infty)$
• D. $(-\infty, \infty)$

Hint:

An absolute value flips any negative input to its positive twin. So, if you feed a negative number into $x + |x|$, they will cancel out perfectly to zero and cause a division-by-zero crash. The function can only breathe when $x$ is strictly positive!

Answer 1

The Correct Option is C

Step 1: Spot the conditions.
We have $f(x)=\frac{1}{\sqrt{x+|x|}}$. The square root needs its inside to be at least 0, and the denominator cannot be 0. Together, we need $x+|x|>0$.

Step 2: Test negative $x$.
If $x<0$, then $|x|=-x$, so $x+|x|=x-x=0$. That fails the strict greater-than-zero rule.

Step 3: Test $x=0$ and positive $x$.
At $x=0$ we again get $0$, which fails. If $x>0$, then $|x|=x$, so $x+|x|=2x>0$, which works.

Step 4: State the domain.
Only positive numbers work. \[ \boxed{(0,\infty)} \]