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The problem provides two functions \( f(x) = 5x + 3 \) and \( g(x) = \frac{1}{x} \). You are asked to compute the following compositions of functions:

1. \( (f \circ g)(1) \)
2. \( (g \circ f)(-2) \)

Let's solve these step by step.

### Part (a) \( (f \circ g)(1) \)
The composition \( (f \circ g)(1) \) means you first apply \( g(x) \), and then apply \( f(x) \) to the result of \( g(1) \).

1. Calculate \( g(1) \):
   \[
   g(1) = \frac{1}{1} = 1
   \]
2. Now, calculate \( f(g(1)) = f(1) \):
   \[
   f(1) = 5(1) + 3 = 5 + 3 = 8
   \]
So, \( (f \circ g)(1) = 8 \).

### Part (b) \( (g \circ f)(-2) \)
The composition \( (g \circ f)(-2) \) means you first apply \( f(x) \), and then apply \( g(x) \) to the result of \( f(-2) \).

1. Calculate \( f(-2) \):
   \[
   f(-2) = 5(-2) + 3 = -10 + 3 = -7
   \]
2. Now, calculate \( g(f(-2)) = g(-7) \):
   \[
   g(-7) = \frac{1}{-7} = -\frac{1}{7}
   \]
So, \( (g \circ f)(-2) = -\frac{1}{7} \).

### Final Answers:
1. \( (f \circ g)(1) = 8 \)
2. \( (g \circ f)(-2) = -\frac{1}{7} \)

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

Here are 5 relative questions that expand on this information:
1. How do you interpret the composition of functions in terms of input and output?
2. What would be \( (g \circ g)(x) \) for the functions given?
3. How would you find the inverse of \( f(x) = 5x + 3 \)?
4. What is the domain of the function \( g(x) = \frac{1}{x} \)?
5. Can the composition \( (f \circ f)(x) \) be calculated? If so, how?

**Tip:** Always remember that function composition is not commutative, meaning \( f \circ g \) is generally not the same as \( g \circ f \).

If f(x) = 5x + 3 and g(x) = 1/x, compute (a) (f ∘ g)(1) and (b) (g ∘ f)(-2).

Learn how to solve the composition of functions problem where f(x) = 5x + 3 and g(x) = 1/x. Step-by-step solution for (f ∘ g)(1) and (g ∘ f)(-2) provided, covering key concepts in algebra and function composition. Suitable for students in Grades 9-11.

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