Question

the range of which function includes -4?
○ $y = \sqrt{x} - 5$
○ $y = \sqrt{x} + 5$
○ $y = \sqrt{x + 5}$
○ $y = \sqrt{x - 5}$

Explanation:

Step1: Recall the range of square root function

The square root function \( y = \sqrt{u} \) (where \( u\geq0 \)) has a range of \( y\geq0 \) because the square root of a non - negative number is always non - negative.

Step2: Analyze the range of each function

• For \( y=\sqrt{x}-5 \):

Since \( \sqrt{x}\geq0 \) (for \( x\geq0 \)), then \( y = \sqrt{x}-5\geq0 - 5=-5 \). So the range of this function is \( y\geq - 5 \). We need to check if - 4 is in this range. Since \( - 4\geq - 5 \), - 4 is in the range of \( y=\sqrt{x}-5 \).

• For \( y=\sqrt{x}+5 \):

Since \( \sqrt{x}\geq0 \), then \( y=\sqrt{x}+5\geq0 + 5 = 5 \). The range is \( y\geq5 \), and \( - 4<5 \), so - 4 is not in the range.

• For \( y=\sqrt{x + 5}\):

Since \( x+5\geq0\) (i.e., \( x\geq - 5\)) and \( \sqrt{x + 5}\geq0 \), the range of this function is \( y\geq0 \). \( - 4<0 \), so - 4 is not in the range.

• For \( y=\sqrt{x - 5}\):

Since \( x-5\geq0\) (i.e., \( x\geq5\)) and \( \sqrt{x - 5}\geq0 \), the range of this function is \( y\geq0 \). \( - 4<0 \), so - 4 is not in the range.

Answer:

\( y=\sqrt{x}-5 \) (the first option)