Let's solve the given problem step-by-step.
We have two functions defined as:
1. [tex]\( f(x) = 5x + 7 \)[/tex]
2. [tex]\( g(x) = -2x - 4 \)[/tex]
We need to find the composition [tex]\( g(f(x)) \)[/tex]. Let's break this down.
Step 1: Compute [tex]\( f(x) \)[/tex].
Given [tex]\( f(x) = 5x + 7 \)[/tex].
Step 2: Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex].
We need to find [tex]\( g(f(x)) \)[/tex]. From the definition of [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = -2x - 4 \][/tex]
Replace [tex]\( x \)[/tex] in [tex]\( g(x) \)[/tex] with [tex]\( f(x) \)[/tex]:
[tex]\[ g(f(x)) = g(5x + 7) \][/tex]
Step 3: Evaluate [tex]\( g(f(x)) \)[/tex].
Substitute [tex]\( 5x + 7 \)[/tex] for [tex]\( x \)[/tex] in [tex]\( g(x) \)[/tex]:
[tex]\[ g(5x + 7) = -2(5x + 7) - 4 \][/tex]
Step 4: Simplify the expression.
First, distribute [tex]\(-2\)[/tex] through the parentheses:
[tex]\[ -2(5x + 7) = -10x - 14 \][/tex]
Then, subtract 4:
[tex]\[ -10x - 14 - 4 = -10x - 18 \][/tex]
So, the final form of the composite function [tex]\( g(f(x)) \)[/tex] is:
[tex]\[ g(f(x)) = -10x - 18 \][/tex]
Thus, the coefficient of [tex]\( x \)[/tex] in the expression [tex]\( g(f(x)) \)[/tex] is [tex]\(-10\)[/tex], and the constant term is [tex]\(-18\)[/tex]. Therefore, the missing term in [tex]\( g(f(x)) = -10x + [?] \)[/tex] is:
[tex]\[ \boxed{-18} \][/tex]
