Question

the function $f(x) = \sqrt{-x}$ is shown on the graph.
which statement is correct?
\bigcirc the domain of the function is all real numbers greater than or equal to 0.
\bigcirc the range of the function is all real numbers greater than or equal to -1.
\bigcirc the range of the function is all real numbers less than or equal to 0.
\bigcirc the domain of the function is all real numbers less than or equal to 0.

Explanation:

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

For the square root function \( \sqrt{u} \), the expression inside the square root (\( u \)) must be non - negative. So, for \( f(x)=\sqrt{-x} \), we need \( -x\geq0 \). Multiplying both sides by - 1 (and reversing the inequality sign), we get \( x\leq0 \). So the domain is all real numbers less than or equal to 0.

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

The square root function \( \sqrt{u} \) has a range of \( y\geq0 \) when \( u\geq0 \). In \( f(x)=\sqrt{-x} \), when \( x = 0 \), \( f(0)=\sqrt{0}=0 \), and as \( x \) decreases (becomes more negative), \( -x \) increases, and \( \sqrt{-x} \) also increases. So the range of \( f(x)=\sqrt{-x} \) is all real numbers greater than or equal to 0.
Now let's check each option:

• Option 1: The domain is not all real numbers greater than or equal to 0 (we found domain is \( x\leq0 \)), so this is wrong.
• Option 2: The range is not all real numbers greater than or equal to - 1 (range is \( y\geq0 \)), so this is wrong.
• Option 3: The range is not all real numbers less than or equal to 0 (range is \( y\geq0 \)), so this is wrong.
• Option 4: The domain is all real numbers less than or equal to 0, which matches our calculation.

Answer:

The domain of the function is all real numbers less than or equal to 0. (The option corresponding to this statement)