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

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

Enter:


Sagot :

Sure! Let's go through the steps to find [tex]\( g(f(x)) \)[/tex] given the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].

1. First, substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:

We are given:
[tex]\[ f(x) = x - 7 \][/tex]
[tex]\[ g(x) = 5x + 2 \][/tex]

2. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:

Substitute [tex]\( x - 7 \)[/tex] for [tex]\( x \)[/tex] in [tex]\( g(x) \)[/tex]:
[tex]\[ g(f(x)) = g(x - 7) \][/tex]

3. Evaluate [tex]\( g(x - 7) \)[/tex]:

[tex]\[ g(x - 7) = 5(x - 7) + 2 \][/tex]
Distribute the 5 inside the parentheses:
[tex]\[ g(x - 7) = 5x - 35 + 2 \][/tex]
Combine like terms:
[tex]\[ g(x - 7) = 5x - 33 \][/tex]

4. Identify the coefficients:

We see that:
[tex]\[ g(f(x)) = 5x - 33 \][/tex]

Therefore, [tex]\( g(f(x)) \)[/tex] can be written as:
[tex]\[ g(f(x)) = 5 \cdot x + (-33) \][/tex]

So, the coefficients are 5 and -33 respectively.

Therefore:
[tex]\[ g(f(x)) = 5x - 33 \][/tex]

Hence, the coefficients in the form [tex]\([?\: , \: \square]\)[/tex] are [tex]\([5\: , \: -33]\)[/tex].

Thus,
[tex]\[ g(f(x)) = 5x - 33 \][/tex]
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