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Find the composite functions f(g(x)) and g(f(x)) for the given functions (x)=2x3³and g(x)=3x.

User Gemita
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Final Answer:

The composite functions f(g(x)) and g(f(x)) for the given functions (x)=2x3³and g(x)=3x are
\[ f(g(x)) = 18x^3 \quad \text{and} \quad g(f(x)) = 54x^9 \]

Step-by-step explanation:

In the composite function f(g(x)), you substitute the expression for g(x) into f(x), yielding f(g(x)) =
2(3x)^3. Simplifying, you get f(g(x)) =
18x^3. Conversely, for g(f(x)), you substitute f(x) into g(x), resulting in g(f(x)) =
3(2x^3)^3, which simplifies to g(f(x)) =
54x^9.

The first composite function,
\( f(g(x)) = 18x^3 \), represents the cube of three times the input, while the second composite function,
\( g(f(x)) = 54x^9 \), signifies the ninth power of twice the input.

These composite functions demonstrate the transformative nature of function composition, where the output of one function becomes the input for the other.

User Chandani
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