Question

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

Explanation:

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

For a square root function \( \sqrt{u} \), the expression inside the square root (\( u \)) must be non - negative (i.e., \( u\geq0 \)). Here, \( u = -x \), so we set up the inequality \( -x\geq0 \).
To solve \( -x\geq0 \), we multiply both sides by - 1. Remember that when we multiply or divide an inequality by a negative number, the direction of the inequality sign changes. So, \( x\leq0 \). This means the domain of the function \( f(x)=\sqrt{-x} \) is all real numbers less than or equal to 0.

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

The square root function \( \sqrt{u} \) (where \( u\geq0 \)) always gives a non - negative output. Let \( u=-x \), and since \( u\geq0 \), \( \sqrt{-x}\geq0 \). So the range of the function is all real numbers greater than or equal to 0.

Now let's check each option:

• Option 1: "The domain of the graph is all real numbers." This is false because we found the domain is \( x\leq0 \).
• Option 2: "The range of the graph is all real numbers." This is false because the range is \( y\geq0 \).
• Option 3: "The domain of the graph is all real numbers less than or equal to 0." This is true based on our domain analysis.
• Option 4: "The range of the graph is all real numbers less than or equal to 0." This is false because the range is \( y\geq0 \).

Answer:

C. The domain of the graph is all real numbers less than or equal to 0.