To determine which function defines [tex]\((g \cdot f)(x)\)[/tex], let's analyze the given functions and the composite function step by step.
We are given:
- [tex]\( f(x) = \log(5x) \)[/tex]
- [tex]\( g(x) = 5x + 4 \)[/tex]
The composite function [tex]\((g \cdot f)(x)\)[/tex] means that we first apply [tex]\( f(x) \)[/tex] and then apply [tex]\( g \)[/tex] to the result of [tex]\( f(x) \)[/tex]. In other words, [tex]\((g \cdot f)(x) = g(f(x))\)[/tex].
First, we calculate [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = \log(5x) \][/tex]
Next, we apply [tex]\( g \)[/tex] to [tex]\( f(x) \)[/tex]:
Let [tex]\( y = f(x) = \log(5x) \)[/tex].
Now, substitute [tex]\( y \)[/tex] into the function [tex]\( g(y) \)[/tex]:
[tex]\[ g(y) = g(\log(5x)) \][/tex]
[tex]\[ g(\log(5x)) = 5 \log(5x) + 4 \][/tex]
Thus, the composite function [tex]\( (g \cdot f)(x) \)[/tex] is:
[tex]\[ (g \cdot f)(x) = 5 \log(5x) + 4 \][/tex]
Therefore, the correct answer is:
C. [tex]\((g \cdot f)(x) = 5 \log(5x) + 4\)[/tex]
