If 𝑓(𝑔(𝑥))=cos(5𝑥+4), find the functions 𝑓(𝑥) and 𝑔(𝑥).Note: there may be more than one solution, but do not use the trivial solution 𝑔(𝑥)=𝑥.𝑓(𝑥)= 𝑔(𝑥)=

To find the functions $f(x)$ and $g(x)$ such that $f(g(x)) = \cos(5x + 4)$, we can explore the composition of functions and the structure of the cosine function.

Step 1: Analyzing the Composition

The given equation states that we need to perform function composition. In this case, we have $f(g(x)) = \cos(5x + 4)$. We can identify potential functions $g(x)$ and $f(x)$ by manipulating the expression inside the cosine function.

Step 2: Choosing a Suitable $g(x)$

A reasonable choice for $g(x)$ could be a linear transformation. For example, let us define: $g(x) = 5x + 4$ This will transform $x$ into the argument of the cosine function.

Step 3: Defining $f(x)$

With our choice of $g(x)$, we then need to find $f(x)$ such that: $f(5x + 4) = \cos(5x + 4)$ This leads us to define $f(x)$ in terms of its argument. Thus, we can let: $f(x) = \cos(x)$

Final Answer

The functions are: $g(x) = 5x + 4$ $f(x) = \cos(x)$

This satisfies the given relation, and it is not a trivial solution since $g(x)$ does not equal $x$. Other non-trivial combinations of $f$ and $g$ could also exist, but this pair is a simple and valid solution.