Question

the square root function
which statement is true about the function ( f(x) = sqrt{x} )?

• the domain of the graph is all real numbers.
• the range of the graph is all real numbers.
• the domain of the graph is all real numbers less than or equal to 0.
• the range of the graph is all real numbers greater than or equal to 0.

Explanation:

Step1: Recall domain/range of square root function

For \( f(x)=\sqrt{x} \), the expression under the square root (radicand) must be non - negative. So, \( x\geq0 \), which means the domain is all real numbers \( x \) such that \( x\geq0 \).

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

The square root of a non - negative number is always non - negative. That is, if \( y = \sqrt{x} \), then \( y\geq0 \) for all \( x\geq0 \) in the domain. So the range of \( f(x)=\sqrt{x} \) is all real numbers greater than or equal to 0.

Step3: Evaluate each option

• Option 1: "The domain of the graph is all real numbers." - False, since \( x\) must be non - negative.
• Option 2: "The range of the graph is all real numbers." - False, since the square root function outputs non - negative values.
• Option 3: "The domain of the graph is all real numbers less than or equal to 0." - False, because we need \( x\geq0 \) for \( \sqrt{x} \) to be a real number.
• Option 4: "The range of the graph is all real numbers greater than or equal to 0." - True, as explained in Step 2.

Answer:

The statement "The range of the graph is all real numbers greater than or equal to 0." (the fourth option) is true about the function \( f(x)=\sqrt{x} \).