To complete the table, let's break down the function [tex]\( y = f(g(x)) \)[/tex] into its individual components [tex]\( g(x) \)[/tex] and [tex]\( f(u) \)[/tex].
1. We are given [tex]\( y = \sin\left(\frac{4x}{7}\right) \)[/tex].
2. We need to identify [tex]\( g(x) \)[/tex]:
- Notice that inside the sine function, we have [tex]\( \frac{4x}{7} \)[/tex].
- Therefore, we can define [tex]\( g(x) = \frac{4x}{7} \)[/tex].
3. Next, we need to identify [tex]\( f(u) \)[/tex]:
- By substituting [tex]\( u = g(x) \)[/tex] into the original function, we have [tex]\( y = \sin(u) \)[/tex].
So, we decompose the given function as follows:
- [tex]\( u = g(x) = \frac{4x}{7} \)[/tex]
- [tex]\( y = f(u) = \sin(u) \)[/tex]
Thus, the completed table will look like this:
\begin{tabular}{|c|c|c|}
\hline
[tex]\( y = f(g(x)) \)[/tex] & [tex]\( u = g(x) \)[/tex] & [tex]\( y = f(u) \)[/tex] \\
\hline
[tex]\( y = \sin\left(\frac{4x}{7}\right) \)[/tex] & [tex]\( u = \frac{4x}{7} \)[/tex] & [tex]\( y = \sin(u) \)[/tex] \\
\hline
\end{tabular}
