Question

what are the domain and range of this function?
y = \sqrt{2x + 16} - 8

domain: {x|x \geq 8}, range: {y|y \geq -8}
domain: {x|x \geq -8}, range: {y|y \geq -8}
domain: {x|x \geq 8}, range: {y|y \geq 2}
domain: {x|x \geq -8}, range: {y|y \geq 2}

Explanation:

Step1: Find the domain of the function

For the square root function \( y = \sqrt{2x + 16}-8 \), the expression inside the square root must be non - negative. So we set up the inequality:
\( 2x+16\geq0 \)
Subtract 16 from both sides: \( 2x\geq - 16 \)
Divide both sides by 2: \( x\geq - 8 \)
So the domain is \( \{x|x\geq - 8\} \)

Step2: Find the range of the function

The square root function \( \sqrt{2x + 16} \) has a range of \( \{y|y\geq0\} \) because the square root of a non - negative number is always non - negative.
For the function \( y=\sqrt{2x + 16}-8 \), we subtract 8 from the square root part.
If \( \sqrt{2x + 16}\geq0 \), then \( \sqrt{2x + 16}-8\geq0 - 8=-8 \)
So the range of the function \( y = \sqrt{2x + 16}-8 \) is \( \{y|y\geq - 8\} \)

Answer:

domain: \(\{x|x\geq - 8\}\), range: \(\{y|y\geq - 8\}\) (corresponding to the third option)