Question

which statement is true about the function $f(x) = \sqrt{-x}$?

• it has the same domain as the function $f(x) = -\sqrt{-x}$.
• it has the same range as the function $f(x) = -\sqrt{-x}$.
• it has the same domain as the function $f(x) = -\sqrt{x}$.
• it has the same range as the function $f(x) = -\sqrt{x}$.

Explanation:

Step1: Find domain of \( f(x)=\sqrt{-x} \)

For square root, the expression inside must be non - negative. So, \(-x\geq0\), which implies \(x\leq0\). The range of \(y = \sqrt{-x}\) is \(y\geq0\) (since square root of a non - negative number is non - negative).

Step2: Analyze \( f(x)=-\sqrt{-x} \)

Domain: For \(-\sqrt{-x}\), the expression inside the square root \(-x\geq0\), so \(x\leq0\). Range: Since \(\sqrt{-x}\geq0\), then \(-\sqrt{-x}\leq0\).

Step3: Analyze \( f(x)=-\sqrt{x} \)

Domain: For \(\sqrt{x}\), \(x\geq0\). Range: Since \(\sqrt{x}\geq0\), then \(-\sqrt{x}\leq0\).

Answer:

It has the same domain as the function \( f(x)=-\sqrt{-x} \).