To determine which function defines [tex]\((g \cdot f)(x)\)[/tex], we need to understand what [tex]\( (g \cdot f)(x) \)[/tex] stands for. Given the problem, we have:
[tex]\[ f(x) = \log(5x) \][/tex]
[tex]\[ g(x) = 5x + 4 \][/tex]
The product [tex]\( (g \cdot f)(x) \)[/tex] represents the multiplication of the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ (g \cdot f)(x) = g(x) \cdot f(x) \][/tex]
Now we substitute [tex]\( g(x) \)[/tex] and [tex]\( f(x) \)[/tex] into this expression:
[tex]\[ (g \cdot f)(x) = (5x + 4) \cdot \log(5x) \][/tex]
We want to compare this with the provided options to see which one matches:
A. [tex]\( 5x \log(5x) + 4 \)[/tex]
B. [tex]\( 5x - 4 - \log(5x) \)[/tex]
C. [tex]\( 5x + 4 + \log(5x) \)[/tex]
D. [tex]\( 5x \log(5x) + 4 \log(5x) \)[/tex]
First, observe the basic structure of each option:
- Option A: [tex]\( 5x \log(5x) + 4 \)[/tex]
- Option B: [tex]\( 5x - 4 - \log(5x) \)[/tex]
- Option C: [tex]\( 5x + 4 + \log(5x) \)[/tex]
- Option D: [tex]\( 5x \log(5x) + 4 \log(5x) \)[/tex]
Next, comparing these with [tex]\((5x + 4) \cdot \log(5x)\)[/tex]:
Let's distribute [tex]\(\log(5x)\)[/tex] from the expression:
[tex]\[ (5x + 4) \cdot \log(5x) = 5x \log(5x) + 4 \log(5x) \][/tex]
From this, it is clear that the expression expands to:
[tex]\[ 5x \log(5x) + 4 \log(5x) \][/tex]
Upon comparing the correctly expanded expression with the provided options, the matching option is:
D. [tex]\( 5x \log(5x) + 4 \log(5x) \)[/tex]
Therefore, the correct function that defines [tex]\( (g \cdot f)(x) \)[/tex] is:
Option D: [tex]\( (g \cdot f)(x) = 5x \log(5x) + 4 \log(5x) \)[/tex]
