AP DSC School Assistant Math Model Test 3

Concept:The function $f(x) = \sin(\sin x)$ is a composition of sine with itself. The inner sine function $\sin x$ always lies between $-1$ and $1$. Therefore, the outer sine takes inputs only in $[-1, 1]$, and its maximum and minimum occur when the inner sine is at its own extremes.Explanation:We know that for any real $x$, the range of $\sin x$ is $[-1, 1]$. So $\sin x$ attains its maximum value $1$ at $x = \frac{\pi}{2} + 2k\pi$ and its minimum value $-1$ at $x = -\frac{\pi}{2} + 2k\pi$. Thus, the input to the outer $\sin$ will be either $1$ or $-1$ at those points. - When $\sin x = 1$, we get $f(x) = \sin(1)$, which is the maximum possible value because $\sin$ is increasing on $[-1, 1]$.- When $\sin x = -1$, we get $f(x) = \sin(-1)$, which is the minimum possible value.No other input gives a larger or smaller output. Hence the maximum is $\sin 1$ and the minimum is $\sin(-1)$.Answer:Option B: $\sin(1)$ and $\sin(-1)$.