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Using the following equations, find [tex]g(f(x))[/tex]:

[tex]\[
\begin{array}{c}
f(x) = 5x + 7 \\
g(x) = -2x - 4 \\
\end{array}
\][/tex]

Calculate [tex]g(f(x))[/tex].

Sagot :

GinnyAnswer
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]
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