To determine the function [tex]\((g \cdot f)(x)\)[/tex], we need to find [tex]\(g(f(x))\)[/tex]. This means we will substitute [tex]\(f(x)\)[/tex] into [tex]\(g(x)\)[/tex].
We are given:
[tex]\[ f(x) = \log(5x) \][/tex]
[tex]\[ g(x) = 5x + 4 \][/tex]
Since we need [tex]\(g(f(x))\)[/tex], we substitute [tex]\(f(x)\)[/tex] into [tex]\(g(x)\)[/tex]:
1. Substitute [tex]\(f(x)\)[/tex] into [tex]\(g(x)\)[/tex]:
[tex]\[
g(f(x)) = g(\log(5x))
\][/tex]
2. Replace [tex]\(x\)[/tex] in [tex]\(g(x) = 5x + 4\)[/tex] with [tex]\(\log(5x)\)[/tex]:
[tex]\[
g(\log(5x)) = 5 \cdot \log(5x) + 4
\][/tex]
Therefore, the function [tex]\((g \cdot f)(x)\)[/tex] simplifies to:
[tex]\[
(g \cdot f)(x) = 5 \log(5x) + 4
\][/tex]
Now we need to match this expression with the given options:
A. [tex]\((g \cdot f)(x) = 5x + 4 + \log(5x)\)[/tex]
B. [tex]\((g \cdot f)(x) = 5x \log(5x) + 4 \log(5x)\)[/tex]
C. [tex]\((g \cdot f)(x) = 5x \log(5x) + 4\)[/tex]
D. [tex]\((g \cdot f)(x) = 5x - 4 - \log(5x)\)[/tex]
After comparing, the correct formula from our expression [tex]\(5 \log(5x) + 4\)[/tex] is:
C. [tex]\((g \cdot f)(x) = 5x \log(5x) + 4\)[/tex]
Thus, the correct answer is:
C. [tex]\((g \cdot f)(x) = 5x \log(5x) + 4\)[/tex]
