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Decompose the function $f(g(h(x)))=3^{(\cos x+6)}$ into $f(x)$ , $g(x)$ and $h(x)$ .

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Find the vertex of the transformed function

Full solution

Q. Decompose the function

f(g(h(x)))=3^{(\cos x+6)}

into

f(x)

,

g(x)

and

h(x)

.

Identify Functions: To decompose the function $f(g(h(x))) = 3^{\cos x + 6}$ , we need to identify the innermost function, the middle function, and the outermost function.
Innermost Function: The innermost function is $h(x)$ , which is the argument of the cosine function. Since the cosine function is $\cos(x)$ , we can deduce that $h(x) = x$ .
Middle Function: The middle function is $g(x)$ , which is the argument of the exponential function. Since the exponential function is $3^{(\cos x + 6)}$ , and we have already identified that $h(x) = x$ , we can deduce that $g(x) = \cos(h(x)) + 6 = \cos(x) + 6$ .
Outermost Function: The outermost function is $f(x)$ , which is the exponential function. Since the base of the exponential function is $3$ and the exponent is the result of $g(x)$ , we can deduce that $f(x) = 3^x$ .

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