To solve for [tex]\((g \circ f)(3)\)[/tex], we need to follow these steps:
1. Find [tex]\( f(3) \)[/tex]:
- We are given the function [tex]\( f(x) = 4x + 2 \)[/tex].
- Substitute [tex]\( x = 3 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(3) = 4(3) + 2 = 12 + 2 = 14
\][/tex]
2. Now, use the result of [tex]\( f(3) \)[/tex] to find [tex]\( g(f(3)) \)[/tex]:
- We found [tex]\( f(3) = 14 \)[/tex].
- Next, we are given the function [tex]\( g(x) = x^2 - 2x - 5 \)[/tex].
- Substitute [tex]\( x = 14 \)[/tex] into the function [tex]\( g(x) \)[/tex]:
[tex]\[
g(14) = 14^2 - 2(14) - 5
\][/tex]
3. Calculate [tex]\( g(14) \)[/tex]:
- First, compute [tex]\( 14^2 \)[/tex]:
[tex]\[
14^2 = 196
\][/tex]
- Then, compute [tex]\( 2(14) \)[/tex]:
[tex]\[
2(14) = 28
\][/tex]
- Now, substitute these values back into [tex]\( g(14) \)[/tex]:
[tex]\[
g(14) = 196 - 28 - 5 = 168 - 5 = 163
\][/tex]
Therefore, the value of [tex]\((g \circ f)(3)\)[/tex] is [tex]\(163\)[/tex].
