Question

The domain of the function $f\left(x\right)=\sqrt{x-x^2}+\sqrt{4+x}+\sqrt{4-x}$ is ?

• A. $[-4,\infty )$
• B. $[-4,4]$
• C. $[0,4]$
• D. $[0,1]$

Answer 1

Answer: (D)

$[0,1]$

To find the domain, all the square roots must be properly defined.
For square-root to be properly defined, the expression inside it must be non-negative.
Therefore:
$x-x^2\ge 0$ (1)
$4+x\ge 0$ (2)
$4-x\ge 0$ (3)
Solving (1):
$x-x^2\ge 0$
$⟹x(1-x)\ge 0$
$⟹x(x-1)\le 0$
$x\in [0,1]$ (i)
Solving (2):
$4+x\ge 0$
$⟹x\ge -4$
$x\in [-4,\infty )$ (ii)
Solving (3):
$4-x\ge 0$
$⟹x\le 4$
$x\in (-\infty ,4]$ (iii)
For all to be properly defined at once, take the intersection of (i), (ii) and (iii):
$x\in [0,1]$