Question

the square root function
quiz complete
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the function f(x)=√(-x) is shown on the graph. which statement is correct?
the range of the graph is all real numbers less than or equal to 0.
the domain of the graph is all real numbers less than or equal to 0.
the domain and range of the graph are the same.
the range of the graph is all real numbers.
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Explanation:

1. First, analyze the domain of \( f(x)=\sqrt{-x} \). For the square root function, the expression inside the square root (the radicand) must be non - negative. So, we set up the inequality:

• The radicand is \(-x\), and we need \(-x\geq0\). Solving the inequality \(-x\geq0\) by multiplying both sides by \(- 1\) (and reversing the inequality sign), we get \(x\leq0\). So the domain of the function is all real numbers less than or equal to 0.

1. Next, analyze the range of \( f(x)=\sqrt{-x} \). Let \(t = - x\), where \(t\geq0\) (from the domain analysis). Then \(y=\sqrt{t}\), and since \(t\geq0\), \(\sqrt{t}\geq0\). So the range of the function is all real numbers greater than or equal to 0.
2. Now, let's check each option:

• Option 1: The range of the graph is all real numbers less than or equal to 0. This is incorrect because the range is \(y\geq0\).
• Option 2: The domain of the graph is all real numbers less than or equal to 0. This is correct as we found from the radicand analysis.
• Option 3: The domain and range of the graph are the same. 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 of the graph is all real numbers. This is incorrect because the range is \(y\geq0\), not all real numbers.

Answer:

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