In order to find the function [tex]\((g \cdot f)(x)\)[/tex], we first need to clarify what this notation means. The notation [tex]\((g \cdot f)(x)\)[/tex] typically stands for the composition of the functions [tex]\(g(x)\)[/tex] and [tex]\(f(x)\)[/tex], represented as [tex]\(g(f(x))\)[/tex]. This means we will substitute the entire function [tex]\(f(x)\)[/tex] into the function [tex]\(g(x)\)[/tex].
Given the functions:
[tex]\[
f(x) = \log(5x)
\][/tex]
[tex]\[
g(x) = 5x + 4
\][/tex]
Step-by-step solution:
1. Evaluate [tex]\(f(x)\)[/tex] to get [tex]\(f(x) = \log(5x)\)[/tex].
2. Substitute [tex]\(f(x) = \log(5x)\)[/tex] into [tex]\(g(x)\)[/tex]:
[tex]\[
g(f(x)) = g(\log(5x))
\][/tex]
3. Next, replace [tex]\(x\)[/tex] in [tex]\(g(x)\)[/tex] with [tex]\(\log(5x)\)[/tex]:
[tex]\[
g(\log(5x)) = 5(\log(5x)) + 4
\][/tex]
4. So, the function [tex]\((g \cdot f)(x)\)[/tex] becomes:
[tex]\[
(g \cdot f)(x) = 5 \log(5x) + 4
\][/tex]
Now, let's check if this matches any of the given options:
- A. [tex]\((g \cdot f)(x) = 5x + 4 + \log(5x)\)[/tex]
- B. [tex]\((g \cdot f)(x) = 5x \log(5x) + 4\)[/tex]
- C. [tex]\((g \cdot f)(x) = 5x - 4 - \log(5x)\)[/tex]
- D. [tex]\((g \cdot f)(x) = 5x \log(5x) + 4 \log(5x)\)[/tex]
The correct answer matches our derived function:
[tex]\[
(g \cdot f)(x) = 5 \log(5x) + 4
\][/tex]
None of the given options match this exactly, indicating a potential error in the problem statement. However, based on the derivation, the correct function should ideally be:
[tex]\[
(g \cdot f)(x) = 5 \log(5x) + 4
\][/tex]
