Let S(x)=x^2, let P(x)=2^x, and let s(x)=sinx . Find each of...

Let $S(x)=x^{2},$ let $P(x)=2^{x},$ and let $s(x)=\sin x .$ Find each of the following. In each case your answer should be a number.
(i) $\quad(S \circ P)(y)$.
(ii) $\quad(S \circ s)(y)$.
(iii) $(S \circ P \circ s)(t)+(s \circ P)(t)$.
(iv) $s\left(t^{3}\right)$.

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Key Concepts

Trigonometric Sine Function

The sine function is a fundamental trigonometric function that maps an angle, measured in radians, to the y-coordinate of the corresponding point on the unit circle. It is periodic and plays an important role in modeling oscillatory behavior. When composed with other functions, it transforms its input by applying these oscillatory properties.

Square Function

The square function takes an input and raises it to the power of 2. It is a basic polynomial function that transforms its input by squaring it. Within compositions, after obtaining a result from another function, the square function highlights the effect of non-linear transformations.

Exponential Function

The exponential function, often expressed as a^x (with a constant base, such as 2), models growth or decay processes and is characterized by its constant rate of change proportional to its current value. In the context of function composition, it provides the intermediate value that is further transformed by subsequent functions.

Function Composition

Function composition involves applying one function to the result of another function. It is denoted by (f ? g)(x) = f(g(x)), meaning that you first evaluate g(x) and then use that as the input to f. This concept is central when dealing with nested functions or chains of functions.

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