Find two nontrivial functions $$ f(x) $$ and $$ g(x) $$ so $$ f(g(x))=(-5-9 x)^{4} $$ $$ f(x)=\square $$ $$ g(x)=\square $$

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$$ f(x) = x^4 $$ $$ g(x) = -5 - 9x $$ To find two nontrivial functions $$ f(x) $$ and $$ g(x) $$ such that $$ f(g(x)) = (-5 - 9x)^4 $$, we can start by considering the structure of the expression on the right-hand side. We can choose $$ g(x) $$ to be the inner function that simplifies the expression. A natural choice is to let: $$ g(x) = -5 - 9x $$ This choice makes $$ g(x) $$ a linear function that directly relates to the expression we want to raise to the fourth power. Next, we need to define $$ f(x) $$ such that when we apply it to $$ g(x) $$, we get the desired result: $$ f(g(x)) = f(-5 - 9x) $$ To achieve $$ f(g(x)) = (-5 - 9x)^4 $$, we can define $$ f(x) $$ as: $$ f(x) = x^4 $$ Now, we can check our functions: 1. $$ g(x) = -5 - 9x $$ 2. $$ f(x) = x^4 $$ Now, substituting $$ g(x) $$ into $$ f(x) $$: $$ f(g(x)) = f(-5 - 9x) = (-5 - 9x)^4 $$ This confirms that our choice of functions satisfies the original equation. Therefore, the nontrivial functions are: $$ f(x) = x^4 $$ $$ g(x) = -5 - 9x $$

Find two nontrivial functions \( f(x) \) and \( g(x) \) so \( f(g(x))=(-5-9 x)^{4} \) \( f(x)=\square \) \( g(x)=\square \)

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Find two nontrivial functions \( f(x) \) and \( g(x) \) so \( f(g(x))=(-5-9 x)^{4} \) \( f(x)=\square \) \( g(x)=\square \)

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