If f(x) = 
11
6x + 5
 and g(x) = 

2x + 7
, compute the compositions (f ∘ g)(1) and (g ∘ f)(1).

Let's compute the compositions \( (f \circ g)(1) \) and \( (g \circ f)(1) \) step by step, using the given functions:

- \( f(x) = \frac{11}{6x + 5} \)
- \( g(x) = 2x + 7 \)

### 1. Compute \( (f \circ g)(1) \)
The composition \( (f \circ g)(1) \) means we first apply \( g(1) \), then apply \( f \) to the result.

1. **Find \( g(1) \)**:
   \[
   g(1) = 2(1) + 7 = 2 + 7 = 9
   \]
2. **Now apply \( f \) to \( g(1) = 9 \)**:
   \[
   f(9) = \frac{11}{6(9) + 5} = \frac{11}{54 + 5} = \frac{11}{59}
   \]
Thus, \( (f \circ g)(1) = \frac{11}{59} \).

### 2. Compute \( (g \circ f)(1) \)
The composition \( (g \circ f)(1) \) means we first apply \( f(1) \), then apply \( g \) to the result.

1. **Find \( f(1) \)**:
   \[
   f(1) = \frac{11}{6(1) + 5} = \frac{11}{6 + 5} = \frac{11}{11} = 1
   \]
2. **Now apply \( g \) to \( f(1) = 1 \)**:
   \[
   g(1) = 2(1) + 7 = 2 + 7 = 9
   \]
Thus, \( (g \circ f)(1) = 9 \).

### Final Answer:
- \( (f \circ g)(1) = \frac{11}{59} \)
- \( (g \circ f)(1) = 9 \)

Would you like more details on any of the steps? 

Here are 5 related questions:
1. What is the general formula for \( (f \circ g)(x) \)?
2. How does the composition of functions differ from addition or multiplication of functions?
3. Can we reverse the composition order for any two functions?
4. What happens if we change the given functions \( f(x) \) and \( g(x) \)?
5. What would \( (f \circ g)(x) \) and \( (g \circ f)(x) \) look like as general expressions?

**Tip:** Always be cautious about the order of compositions; \( f \circ g \) is not the same as \( g \circ f \).

If f(x) = 11/(6x + 5) and g(x) = 2x + 7, compute the compositions (f ∘ g)(1) and (g ∘ f)(1).

Learn how to compute the function compositions (f ∘ g)(1) and (g ∘ f)(1) using the given functions f(x) = 11/(6x + 5) and g(x) = 2x + 7. This problem involves key algebraic concepts like function composition, and is suitable for students in Grades 10-12.

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