Answer: The answer is [tex]f(x)=3\cos(x)+3.[/tex]
Step-by-step explanation: In the question, we are given a graph and four options and we are to find to which function the graph belongs.
We can see from the graph, the following values are true
[tex]f(0)=6,~~f(\dfrac{\pi}{2})=3,~~f(\pi)=0,~~f(3\dfrac{\pi}{2})=3,~~f(2\pi)=6,~\textup{etc . . .}[/tex]
Now, if we take [tex]f(x)=6\cos(x),~~\textup{then}[/tex]
[tex]f(0)=6,~~f(\dfrac{\pi}{2})=0\neq 3,[/tex] so the graph does not belong to this function.
If we take [tex]f(x)=3\cos(x)+3,~~\textup{then}[/tex]
[tex]f(0)=6,~~f(\dfrac{\pi}{2})=3,~~f(\pi)=0,~~f(3\dfrac{\pi}{2})=3,~~f(2\pi)=6.[/tex]
We see that all the values of this function matches with the values taken from the graph, so this function describes the graph given in the question.
Also, we can check for remaining two functions as follows
[tex]f(0)=6\sin(0)=0\neq 6\\\\\textup{and}~~f(0)=3\sin(0)+3=3\neq 6.[/tex]
Thus, the correct function is
[tex]f(x)=3\cos(x)+3.[/tex]
