The domain of the function \(f(x)=\displaystyle\frac{\sqrt{2x-1}+\sqrt{4x+7}}{\sqrt{4-x}+\sqrt{3+17x}}\) is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\left[\displaystyle\frac{1}{2},4\right]\)
Each square root needs a non-negative expression, and all four conditions must hold together:
\( \displaystyle 2x - 1 \ge 0 \Rightarrow x \ge \frac{1}{2} \)
\( \displaystyle 4x + 7 \ge 0 \Rightarrow x \ge -\frac{7}{4} \)
\( 4 - x \ge 0 \Rightarrow x \le 4 \)
\( \displaystyle 3 + 17x \ge 0 \Rightarrow x \ge -\frac{3}{17} \)
The common part is \(\frac{1}{2} \le x \le 4\).
The denominator is never zero there, because \(\sqrt{3+17x} \gt 0\) for \(x \ge \frac{1}{2}\). So the domain is \(\left[\frac{1}{2}, 4\right]\).