Question

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

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 \( -x\geq0 \), which implies \( x\leq0 \). 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 \( \sqrt{u} \) (where \( u = -x\geq0 \)) has a range of \( y\geq0 \), because the square root of a non - negative number is non - negative.

Step3: Evaluate each option

• Option 1: The range is all real numbers less than or equal to 0. This is incorrect because the range of \( \sqrt{-x} \) is \( y\geq0 \).
• Option 2: The domain of the graph is all real numbers less than or equal to 0. This is correct based on our domain analysis.
• Option 3: The domain is \( x\leq0 \) and the range is \( y\geq0 \), so they are not the same. This option is incorrect.
• Option 4: The range is not all real numbers, it is \( y\geq0 \). This option is incorrect.

Answer:

The domain of the graph is all real numbers less than or equal to 0. (The option corresponding to this statement in the multiple - choice is the second option: "The domain of the graph is all real numbers less than or equal to 0.")