Use the pair of functions to find  f(g(x))  and  g(f(x)) . Simplify your answers.
 
f(x)=1x−5 ,  g(x)=9x+5 
 
f(g(x))=      
g(f(x))=      

To find \( f(g(x)) \) and \( g(f(x)) \), we'll substitute each function into the other.

Given:
\[
f(x) = \frac{1}{x} - 5 \quad \text{and} \quad g(x) = 9x + 5
\]

### Step 1: Find \( f(g(x)) \)
1. Substitute \( g(x) \) into \( f(x) \):
   \[
   f(g(x)) = f(9x + 5)
   \]
2. Since \( f(x) = \frac{1}{x} - 5 \), replace \( x \) in \( f(x) \) with \( 9x + 5 \):
   \[
   f(g(x)) = \frac{1}{9x + 5} - 5
   \]

So,
\[
f(g(x)) = \frac{1}{9x + 5} - 5
\]

### Step 2: Find \( g(f(x)) \)
1. Substitute \( f(x) \) into \( g(x) \):
   \[
   g(f(x)) = g\left(\frac{1}{x} - 5\right)
   \]
2. Since \( g(x) = 9x + 5 \), replace \( x \) in \( g(x) \) with \( \frac{1}{x} - 5 \):
   \[
   g(f(x)) = 9\left(\frac{1}{x} - 5\right) + 5
   \]
3. Distribute \( 9 \):
   \[
   g(f(x)) = \frac{9}{x} - 45 + 5
   \]
4. Combine like terms:
   \[
   g(f(x)) = \frac{9}{x} - 40
   \]

So,
\[
g(f(x)) = \frac{9}{x} - 40
\]

### Final Answers
\[
f(g(x)) = \frac{1}{9x + 5} - 5
\]
\[
g(f(x)) = \frac{9}{x} - 40
\]

Would you like further details or have any questions?

#### Related Questions
1. What are the domain restrictions for \( f(g(x)) \) and \( g(f(x)) \)?
2. How would you find \( g(g(x)) \) or \( f(f(x)) \) using the same functions?
3. What are the inverses of \( f(x) \) and \( g(x) \), if they exist?
4. How would changes in the constants (like \( -5 \) or \( +5 \)) affect the results?
5. Can we compose functions in the reverse order, and how does it affect the outcome?

#### Tip
When composing functions, keep in mind that \( f(g(x)) \) and \( g(f(x)) \) often have different domains based on the inner functions. Always check for any restrictions before simplifying.

Use the pair of functions to find f(g(x)) and g(f(x)). Simplify your answers. f(x) = 1/x - 5, g(x) = 9x + 5

Explore how to compose functions f(x) = 1/x - 5 and g(x) = 9x + 5 by finding f(g(x)) and g(f(x)). This solution demonstrates key algebraic techniques, including function composition and simplifying rational expressions, suited for Grades 10-12.

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