Question

unit 7 review: cube & cube root
determine the domain of ( h(x) = sqrt{-x - 3} + 2 ), as shown in the graph above.
options: ( x leq -3 ), ( x geq -3 ), ( x leq 3 ), ( x geq 3 )

Explanation:

Step1: Analyze the square root function

For the function \( h(x)=\sqrt{-x - 3}+2 \), the expression inside the square root (the radicand) must be non - negative. So we set up the inequality:
\( -x - 3\geq0 \)

Step2: Solve the inequality

Add \( x \) to both sides of the inequality \( -x - 3\geq0 \):
\( - 3\geq x \)
Which is equivalent to \( x\leq - 3 \)

We can also check the graph. The graph of the function has its right - most point at \( x=-3 \) and extends to the left (where \( x \) values are less than - 3). So the domain of the function is all real numbers \( x \) such that \( x\leq - 3 \).

Answer:

\( x\leq - 3 \) (corresponding to the option: \( x\leq - 3 \))