Questions: If f(x)=6x and g(x)=2x^3, find (a) f[g(x)], and (b) g[f(x)]. (a) f[g(x)]= (Simplify your answer.) (b) g[f(x)]= (Simplify your answer.)

Transcript text: If $f(x)=6 x$ and $g(x)=2 x^{3}$, find $(a) f[g(x)]$, and $(b) g[f(x)]$. (a) $f[g(x)]=$ $\square$ (Simplify your answer.) (b) $g[f(x)]=$ $\square$ (Simplify your answer.)

Solution

Solution Steps

Step 1: Find $ f[g(x)] $

To find $ f[g(x)] $, we first substitute $ g(x) = 2x^3 $ into $ f(x) = 6x $:

\[ f[g(x)] = f(2x^3) = 6(2x^3) = 12x^3 \]

Evaluating this at $ x = 1 $:

\[ f[g(1)] = 12(1^3) = 12 \]

Step 2: Find $ g[f(x)] $

Next, we find $ g[f(x)] $ by substituting $ f(x) = 6x $ into $ g(x) = 2x^3 $:

\[ g[f(x)] = g(6x) = 2(6x)^3 = 2 \cdot 216x^3 = 432x^3 \]

Evaluating this at $ x = 1 $:

\[ g[f(1)] = 432(1^3) = 432 \]

Final Answer

Thus, the results are: (a) $ f[g(x)] = 12 $
(b) $ g[f(x)] = 432 $

The final answers are: \[ \boxed{12} \] \[ \boxed{432} \]

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