To find the expression for [tex]\((h \circ g)(-8)\)[/tex], we need to first understand that [tex]\((h \circ g)(x)\)[/tex] means [tex]\(h(g(x))\)[/tex]. Essentially, we are applying the function [tex]\(g(x)\)[/tex] first to the value [tex]\(-8\)[/tex], and then taking the result and applying the function [tex]\(h(x)\)[/tex] to it. Here are the step-by-step details:
1. Step 1: Calculate [tex]\(g(-8)\)[/tex]
The function [tex]\(g(x)\)[/tex] is defined as:
[tex]\[
g(x) = 4x - 2
\][/tex]
Substitute [tex]\(x = -8\)[/tex] into the function [tex]\(g(x)\)[/tex]:
[tex]\[
g(-8) = 4(-8) - 2 = -32 - 2 = -34
\][/tex]
Thus, [tex]\(g(-8) = -34\)[/tex].
2. Step 2: Calculate [tex]\(h(-34)\)[/tex]
Now that we have [tex]\(g(-8) = -34\)[/tex], we need to use this result as the input for the function [tex]\(h(x)\)[/tex]. The function [tex]\(h(x)\)[/tex] is defined as:
[tex]\[
h(x) = \frac{1}{2}x + 6
\][/tex]
Substitute [tex]\(x = -34\)[/tex] into the function [tex]\(h(x)\)[/tex]:
[tex]\[
h(-34) = \frac{1}{2}(-34) + 6 = -17 + 6 = -11
\][/tex]
Therefore, [tex]\(h(-34) = -11\)[/tex].
So, to summarize, the calculation is as follows:
1. [tex]\(g(-8) = -34\)[/tex]
2. [tex]\(h(-34) = -11\)[/tex]
Thus, the expression for [tex]\((h \circ g)(-8)\)[/tex] is:
[tex]\[
(h \circ g)(-8) = -11
\][/tex]
