How do you write k(x)=esin⁡x as a composition of two or more functions?

Question

Georgia Ingram · Accepted Answer

In precalculus-related courses, when we write

f (g (x))

, the notation means to first apply the functiongto the numberxto get the number

g (x)

, then apply the functionfto the number

g (x)

to get

f (g (x))

(so we work "from inside out"). (In some other math courses, the inputs and outputs don't have to be numbers).
For the function

k (x) = e^{\sin x}

, when a number x is substituted, the sine function must be evaluated first to get the number

\sin (x)

. This number then must be made the power of e to get the final output

e^{\sin x}

. Therefore, the sine function must be the "inner" (first applied) function and the exponential function must be the "outer" (second applied) function.
Hence,

k (x) = f (g (x))

when

f (x) = e^{x}

and

g (x) = \sin (x)

ocretz56 · Accepted Answer

Extra Info (ignore this at the moment if it&#039;s confusing): An alternative notation for this is to use the &quot;circle operator&quot; and write k(x)=(f∘g)(x). This can also be written as k=f∘g, which emphasizes that ∘ is a &quot;binary operator&quot; that takes two given functions f and g and creates a new function f∘g. In general, the circle operator ∘ is not commutative (that is, in general, f∘g≠g∘f. For the present example, these are indeed not equal because esin⁡(x)≠sin⁡(ex) as functions. Order of operations matters!

How do you write k(x) = e^sinx as a composition

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