Find $a$, such that the function $f ( x ) = 4 x + \sqrt { a - x ^ { 2 } }$ has the domain $( - 4,4 )$.
Select the correct answer.
A) $a = - 16$
B) $a = \sqrt { 4 }$
C) $a = - \sqrt { 4 }$
D) $a = 16$
E) $a = 4$

Question

Quizplus · Accepted Answer

For the domain of the function, we require that the value inside the square root must be non-negative, i.e., 
$$a-x^2\geq 0\Rightarrow x^2\leq a \Rightarrow -\sqrt{a}\leq x \leq \sqrt{a}$$
Also, the domain of the function is given as $(-4,4)$. Therefore, we must have $-4\leq -\sqrt{a}$ and $\sqrt{a}\leq 4$. Combining these inequalities, we get
$$16\geq a\geq 0$$
Out of all the options, only option D satisfies this condition. Therefore, our answer is:
Answer: D) $a = 16$

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