To find the value of [tex]\( g(-4) \)[/tex] given the piecewise function:
[tex]\[ g(x)=\begin{cases}
\sqrt[3]{x+5}, & x \leq -4 \\
-x^2+11, & x > -4
\end{cases}
\][/tex]
we need to determine which condition [tex]\( x = -4 \)[/tex] falls into. Since [tex]\( -4 \leq -4 \)[/tex] is true, we will use the first piece of the piecewise function:
[tex]\[ g(x)= \sqrt[3]{x+5} \][/tex]
Next, we substitute [tex]\( x = -4 \)[/tex] into the first piece of the function:
[tex]\[ g(-4) = \sqrt[3]{-4 + 5} \][/tex]
Simplifying the expression inside the cube root:
[tex]\[ -4 + 5 = 1 \][/tex]
Therefore:
[tex]\[ g(-4) = \sqrt[3]{1} \][/tex]
Since the cube root of 1 is 1:
[tex]\[ g(-4) = 1 \][/tex]
Hence, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
