To solve the given function composition [tex]\((g \circ h)(-2)\)[/tex], we need to apply the function [tex]\(h\)[/tex] first with the input [tex]\(-2\)[/tex], and then apply the function [tex]\(g\)[/tex] to the result of [tex]\(h(-2)\)[/tex].
Here’s a detailed step-by-step solution:
1. Identify the functions:
[tex]\[
g(x) = -20 - 3x
\][/tex]
[tex]\[
h(x) = \left( \frac{1}{2} \right)^x
\][/tex]
2. Evaluate [tex]\(h(-2)\)[/tex]:
[tex]\[
h(-2) = \left( \frac{1}{2} \right)^{-2}
\][/tex]
Using the properties of exponents, [tex]\(\left( \frac{1}{2} \right)^{-2}\)[/tex] is equal to:
[tex]\[
\left( \frac{1}{2} \right)^{-2} = \left( \frac{1}{2} \right)^{-2} = \left( 2 \right)^2 = 4
\][/tex]
So,
[tex]\[
h(-2) = 4
\][/tex]
3. Evaluate [tex]\(g(h(-2))\)[/tex], which means [tex]\(g(4)\)[/tex]:
Substitute [tex]\(4\)[/tex] into the function [tex]\(g(x)\)[/tex]:
[tex]\[
g(4) = -20 - 3 \cdot 4
\][/tex]
4. Perform the arithmetic operations:
[tex]\[
g(4) = -20 - 12 = -32
\][/tex]
Therefore, [tex]\((g \circ h)(-2) = g(h(-2)) = g(4) = -32\)[/tex].
So, the step-by-step solution yields:
[tex]\[
(g \circ h)(-2) = -32
\][/tex]
The intermediate steps result in:
[tex]\[
h(-2) = 4
\][/tex]
[tex]\[
g(4) = -32
\][/tex]
Thus, the final answer is:
[tex]\[
(g \circ h)(-2) = -32
\][/tex]
