Use the pair of functions to find 
f(g(x))
 and 
g(f(x)).
 Simplify your answers.
f(x) = 
3

x
,     g(x) = 
x + 7
x3
f(g(x))	=	
 
g(f(x))	=

To solve for \( f(g(x)) \) and \( g(f(x)) \), we need to substitute one function into the other and simplify.

Given:
\[
f(x) = 3x, \quad g(x) = \frac{x + 7}{x^3}
\]

### 1. Finding \( f(g(x)) \)

To find \( f(g(x)) \), we substitute \( g(x) \) into \( f(x) \):

\[
f(g(x)) = f\left( \frac{x + 7}{x^3} \right)
\]

Since \( f(x) = 3x \), we replace \( x \) in \( f(x) \) with \( \frac{x + 7}{x^3} \):

\[
f(g(x)) = 3 \left( \frac{x + 7}{x^3} \right)
\]

Simplify this expression:

\[
f(g(x)) = \frac{3(x + 7)}{x^3}
\]

### 2. Finding \( g(f(x)) \)

To find \( g(f(x)) \), we substitute \( f(x) \) into \( g(x) \):

\[
g(f(x)) = g(3x)
\]

Since \( g(x) = \frac{x + 7}{x^3} \), we replace \( x \) in \( g(x) \) with \( 3x \):

\[
g(f(x)) = \frac{3x + 7}{(3x)^3}
\]

Now, simplify the denominator:

\[
g(f(x)) = \frac{3x + 7}{27x^3}
\]

### Final Simplified Results

\[
f(g(x)) = \frac{3(x + 7)}{x^3}, \quad g(f(x)) = \frac{3x + 7}{27x^3}
\]

Would you like more details on these steps, or do you have any questions? 

Here are 5 related questions that might interest you:
1. How do you compute the composition of functions in general?
2. What are some real-world applications of function compositions?
3. How do you determine the domain of a composed function?
4. What is the inverse of a composite function?
5. How do you interpret the graph of a composite function?

**Tip:** Remember, when composing functions, always substitute the inner function completely into the outer function and simplify!

Use the pair of functions to find f(g(x)) and g(f(x)). Simplify your answers. f(x) = 3x, g(x) = (x + 7)/x^3

Learn how to compute the composition of two functions, f(x) = 3x and g(x) = (x + 7)/x^3, and simplify f(g(x)) and g(f(x)) step-by-step. This solution highlights key algebraic concepts in function composition, suitable for high school students in Grades 10-12.

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