Find functions $$ f $$ and $$ g $$ so that $$ f \circ g=H $$. $$ H(x)=(9 x+3)^{5} $$

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$$ g(x) = 9x + 3 $$, $$ f(x) = x^5 $$ To find functions $$ f $$ and $$ g $$ such that $$ f \circ g = H $$, where $$ H(x) = (9x + 3)^5 $$, we can start by breaking down the expression for $$ H(x) $$. We can think of $$ H(x) $$ as a composition of two simpler functions. A natural choice is to let $$ g(x) $$ be the inner function that simplifies the expression inside the parentheses, and let $$ f(x) $$ be the outer function that raises the result to the fifth power. Let's define the functions as follows: 1. Let $$ g(x) = 9x + 3 $$. This function takes $$ x $$ and transforms it into the expression inside the parentheses of $$ H(x) $$. 2. Let $$ f(x) = x^5 $$. This function takes the output of $$ g(x) $$ and raises it to the fifth power. Now, we can check the composition $$ f \circ g $$: $$ f \circ g(x) = f(g(x)) = f(9x + 3) = (9x + 3)^5 $$ This matches our original function $$ H(x) $$. Thus, the functions $$ f $$ and $$ g $$ that satisfy $$ f \circ g = H $$ are: $$ g(x) = 9x + 3 $$ $$ f(x) = x^5 $$

Find functions \( f \) and \( g \) so that \( f \circ g=H \). \( H(x)=(9 x+3)^{5} \)

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Find functions \( f \) and \( g \) so that \( f \circ g=H \). \( H(x)=(9 x+3)^{5} \)

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