Given two functions:

• $f(x) = 3x - 1$
• $g(x) = x^2 + 3x + 2$

Let's find:

a. $(fog)(x)$ or $f(g(x))$

To find $(fog)(x)$, we substitute $g(x)$ into $f(x)$.

1. First, identify $g(x) = x^2 + 3x + 2$.
2. Replace $x$ in $f(x)$ with $g(x)$:
   $f(g(x)) = 3(g(x)) - 1$
3. Substitute $g(x)$:
   $f(g(x)) = 3(x^2 + 3x + 2) - 1$
4. Simplify:
   $f(g(x)) = 3x^2 + 9x + 6 - 1$ $f(g(x)) = 3x^2 + 9x + 5$

So, $(fog)(x) = 3x^2 + 9x + 5$.

b. $(gof)(3)$ or $g(f(3))$

To find $(gof)(3)$, we first calculate $f(3)$ and then substitute it into $g(x)$.

1. Calculate $f(3)$: $f(3) = 3(3) - 1 = 9 - 1 = 8$

2. Substitute $f(3) = 8$ into $g(x)$: $g(8) = 8^2 + 3(8) + 2$ $g(8) = 64 + 24 + 2 = 90$

So, $(gof)(3) = 90$.

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

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Here are 5 related questions to consider:

1. What would $(fog)(2)$ be for the same functions?
2. How would $(gof)(x)$ be derived for these two functions in general?
3. Can we find the inverse of $f(x)$? If so, what is it?
4. What is the domain of $g(x)$ for which $(fog)(x)$ is defined?
5. How does $(fog)(x)$ behave as $x$ approaches infinity?

Tip: When evaluating compositions like $f(g(x))$, always substitute $g(x)$ first and then simplify the expression step-by-step.