Let's tackle the given problem step-by-step.
We need to evaluate two composite functions based on the given functions [tex]\( f(x) = x^2 - 7x \)[/tex] and [tex]\( g(x) = x + 9 \)[/tex].
(a) [tex]\((f \circ g)(-1)\)[/tex]
To find [tex]\((f \circ g)(-1)\)[/tex], we need to evaluate the inner function first and then the outer function:
1. Calculate [tex]\( g(-1) \)[/tex]:
[tex]\[
g(-1) = -1 + 9 = 8
\][/tex]
2. Now, we substitute this result into [tex]\( f(x) \)[/tex]:
[tex]\[
f(g(-1)) = f(8)
\][/tex]
3. Calculate [tex]\( f(8) \)[/tex]:
[tex]\[
f(8) = 8^2 - 7 \cdot 8 = 64 - 56 = 8
\][/tex]
Therefore,
[tex]\[
(f \circ g)(-1) = 8
\][/tex]
(b) [tex]\((g \circ f)(-1)\)[/tex]
To find [tex]\((g \circ f)(-1)\)[/tex], we again need to evaluate the inner function first and then the outer function:
1. Calculate [tex]\( f(-1) \)[/tex]:
[tex]\[
f(-1) = (-1)^2 - 7 \cdot (-1) = 1 + 7 = 8
\][/tex]
2. Now, we substitute this result into [tex]\( g(x) \)[/tex]:
[tex]\[
g(f(-1)) = g(8)
\][/tex]
3. Calculate [tex]\( g(8) \)[/tex]:
[tex]\[
g(8) = 8 + 9 = 17
\][/tex]
Therefore,
[tex]\[
(g \circ f)(-1) = 17
\][/tex]
So, the final results are:
[tex]\[
(a) (f \circ g)(-1) = 8
\][/tex]
[tex]\[
(b) (g \circ f)(-1) = 17
\][/tex]
