Let's go through the exercise step by step.

The exercise provided is:

Exercise:

Find (f∘g)(x) and (g∘f)(x) where:

• $f(x) = 3x + 1$
• $g(x) = 2x - 4$

We will solve both compositions of functions:

1. $(f∘g)(x)$

This means $f(g(x))$, where we substitute $g(x)$ into $f(x)$.

Given:

• $f(x) = 3x + 1$
• $g(x) = 2x - 4$

Let's substitute $g(x)$ into $f(x)$: $f(g(x)) = f(2x - 4) = 3(2x - 4) + 1$ Now simplify: $f(g(x)) = 3(2x - 4) + 1 = 6x - 12 + 1 = 6x - 11$ Thus, $(f∘g)(x) = 6x - 11$.

2. $(g∘f)(x)$

This means $g(f(x))$, where we substitute $f(x)$ into $g(x)$.

Given:

• $f(x) = 3x + 1$
• $g(x) = 2x - 4$

Let's substitute $f(x)$ into $g(x)$: $g(f(x)) = g(3x + 1) = 2(3x + 1) - 4$ Now simplify: $g(f(x)) = 2(3x + 1) - 4 = 6x + 2 - 4 = 6x - 2$ Thus, $(g∘f)(x) = 6x - 2$.

Final Answers:

• $(f∘g)(x) = 6x - 11$
• $(g∘f)(x) = 6x - 2$

Let me know if you'd like more details or explanations about these calculations!

Here are 5 related questions for you to explore:

1. What is the general process for composing two functions?
2. How does the order of function composition affect the outcome?
3. Can the functions $f(x)$ and $g(x)$ be any kind of function, such as non-linear functions?
4. What happens if the inner function in the composition is undefined for certain values of $x$?
5. How would the results change if the functions were defined differently, for example, $f(x) = x^2$ and $g(x) = 3x + 1$?

Tip: When composing functions, always be mindful of the domains of both functions to ensure that the composite function is well-defined.