To find [tex]\((f \circ g)(-3)\)[/tex], we need to first find [tex]\(g(-3)\)[/tex] and then use that result as the input for the function [tex]\(f(x)\)[/tex].
1. Find [tex]\(g(-3)\)[/tex]:
The function [tex]\(g(x)\)[/tex] is given by:
[tex]\[
g(x) = 3 - x
\][/tex]
Substituting [tex]\(-3\)[/tex] for [tex]\(x\)[/tex] in [tex]\(g(x)\)[/tex]:
[tex]\[
g(-3) = 3 - (-3) = 3 + 3 = 6
\][/tex]
So, [tex]\(g(-3) = 6\)[/tex].
2. Find [tex]\(f(g(-3))\)[/tex], which is [tex]\(f(6)\)[/tex]:
The function [tex]\(f(x)\)[/tex] is given by:
[tex]\[
f(x) = x^2 - 20
\][/tex]
Substituting [tex]\(6\)[/tex] for [tex]\(x\)[/tex] in [tex]\(f(x)\)[/tex]:
[tex]\[
f(6) = 6^2 - 20 = 36 - 20 = 16
\][/tex]
So, [tex]\(f(6) = 16\)[/tex].
Therefore, the composition [tex]\((f \circ g)(-3)\)[/tex] is:
[tex]\[
(fg)(-3) = 16
\][/tex]
Thus, [tex]\((f g)(-3) = 16\)[/tex].
