To find [tex]\( a(b(5)) \)[/tex], we need to follow a step-by-step approach. Let's break it down:
1. Evaluate [tex]\( b(5) \)[/tex]:
We start by substituting [tex]\( x = 5 \)[/tex] into the function [tex]\( b(x) \)[/tex].
[tex]\[
b(x) = 3x - 5
\][/tex]
So,
[tex]\[
b(5) = 3 \cdot 5 - 5 = 15 - 5 = 10
\][/tex]
2. Use the result of [tex]\( b(5) \)[/tex] to evaluate [tex]\( a(b(5)) \)[/tex]:
We have found that [tex]\( b(5) = 10 \)[/tex]. Now we need to substitute [tex]\( x = 10 \)[/tex] into the function [tex]\( a(x) \)[/tex].
[tex]\[
a(x) = -x^2 + 7x + 1
\][/tex]
So,
[tex]\[
a(10) = -(10)^2 + 7 \cdot 10 + 1
\][/tex]
Simplifying inside the function:
[tex]\[
a(10) = -100 + 70 + 1 = -100 + 71 = -29
\][/tex]
Hence, the final result is:
[tex]\[
a(b(5)) = -29
\][/tex]
So, [tex]\( b(5) = 10 \)[/tex] and [tex]\( a(b(5)) = -29 \)[/tex].
