IDNLearn.com is the place where your questions are met with thoughtful and precise answers. Get prompt and accurate answers to your questions from our experts who are always ready to help.

Select the correct answer.

Which function defines [tex](g \cdot f)(x)[/tex]?
[tex]\[
\begin{array}{l}
f(x)=\log (5 x) \\
g(x)=5 x+4
\end{array}
\][/tex]

A. [tex](g \cdot f)(x)=5 x \log (5 x)+4[/tex]
B. [tex](g \cdot f)(x)=5 x \log (5 x)+4 \log (5 x)[/tex]
C. [tex](g \cdot f)(x)=5 x-4-\log (5 x)[/tex]
D. [tex](g \cdot f)(x)=5 x+4+\log (5 x)[/tex]


Sagot :

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]