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. $(g \circ f)(-2) = -\frac{1}{7}$

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 $g \circ f$.