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Select the correct answer.

Which function defines [tex](g \cdot f)(x)[/tex]?
[tex]\[
\begin{array}{l}
f(x)=\log (5 x) \\
g(x)=5 x+4
\end{array}
\][/tex]

A. [tex](g \cdot f)(x)=5 x \log (5 x)+4[/tex]
B. [tex](g \cdot f)(x)=5 x \log (5 x)+4 \log (5 x)[/tex]
C. [tex](g \cdot f)(x)=5 x-4-\log (5 x)[/tex]
D. [tex](g \cdot f)(x)=5 x+4+\log (5 x)[/tex]



Answer :

GinnyAnswer
To determine [tex]\((g \cdot f)(x)\)[/tex], we need to find the composition of the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]. Specifically, we first apply [tex]\( f(x) \)[/tex] and then apply [tex]\( g \)[/tex] to the result.

Given:
- [tex]\( f(x) = \log(5x) \)[/tex]
- [tex]\( g(x) = 5x + 4 \)[/tex]

We want to find [tex]\( (g \cdot f)(x) = g(f(x)) \)[/tex].

Step-by-step solution:
1. Compute [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = \log(5x) \][/tex]

2. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(f(x)) = g(\log(5x)) \][/tex]

3. Now, apply the function [tex]\( g \)[/tex] to [tex]\( \log(5x) \)[/tex]:
[tex]\[ g(\log(5x)) = 5 \log(5x) + 4 \][/tex]

Combining these steps, we find:
[tex]\[ (g \cdot f)(x) = 5 \log(5x) + 4 \][/tex]

Therefore, the correct answer is:
[tex]\[ D. 5 \log(5x) + 4 \][/tex]

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