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]
