The problem states:

6. Given: $f(x) = 3x - 2$ and $g(x) = x^2 + 3x - 4$, determine:

a. $(g \circ f)(x)$
b. $(f \circ g)(x)$

Solution:

For function composition, we need to substitute one function into the other.

a. $(g \circ f)(x)$: This means $g(f(x))$.

1. Start with $f(x) = 3x - 2$.
2. Substitute $f(x)$ into $g(x) = x^2 + 3x - 4$:

$g(f(x)) = (3x - 2)^2 + 3(3x - 2) - 4$

Now simplify: $= (9x^2 - 12x + 4) + (9x - 6) - 4$ $= 9x^2 - 12x + 4 + 9x - 6 - 4$ $= 9x^2 - 3x - 6$

So, $(g \circ f)(x) = 9x^2 - 3x - 6$.

b. $(f \circ g)(x)$: This means $f(g(x))$.

1. Start with $g(x) = x^2 + 3x - 4$.
2. Substitute $g(x)$ into $f(x) = 3x - 2$:

$f(g(x)) = 3(x^2 + 3x - 4) - 2$

Now simplify: $= 3x^2 + 9x - 12 - 2$ $= 3x^2 + 9x - 14$

So, $(f \circ g)(x) = 3x^2 + 9x - 14$.

Would you like any further explanation on any step?

Here are 5 related questions for further exploration:

1. How do you graph $g(f(x))$ and $f(g(x))$?
2. What are the domains of $g(f(x))$ and $f(g(x))$?
3. How does the composition of functions differ from addition or multiplication of functions?
4. Can function compositions be commutative? Why or why not?
5. What happens if $f(x)$ or $g(x)$ have restricted domains or ranges?

Tip: Always pay attention to the domain when composing functions, as the result might be restricted depending on the functions involved.