Given the functions $f(x) = 3x^2 + 5x + 7$ and $g(x) = 3x + 4$, we are tasked with finding:

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

This represents the composition of the functions, meaning we need to evaluate $f(g(x))$. To do this, substitute $g(x)$ into the function $f(x)$.

1. Start by recalling $g(x) = 3x + 4$. We substitute this into $f(x)$:

$f(g(x)) = f(3x + 4)$

2. Now, substitute $3x + 4$ wherever $x$ appears in $f(x) = 3x^2 + 5x + 7$:

$f(3x + 4) = 3(3x + 4)^2 + 5(3x + 4) + 7$

3. First, expand $(3x + 4)^2$:

$(3x + 4)^2 = 9x^2 + 24x + 16$

4. Substitute this back into $f(3x + 4)$:

$f(3x + 4) = 3(9x^2 + 24x + 16) + 5(3x + 4) + 7$

5. Now, distribute the constants:

$f(3x + 4) = 27x^2 + 72x + 48 + 15x + 20 + 7$

6. Combine like terms:

$f(3x + 4) = 27x^2 + (72x + 15x) + (48 + 20 + 7)$

$f(3x + 4) = 27x^2 + 87x + 75$

Thus, the composition $(f \circ g)(x)$ is:

$(f \circ g)(x) = 27x^2 + 87x + 75$

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

This represents the composition $g(f(x))$. To do this, substitute $f(x)$ into the function $g(x)$.

1. Start by recalling $f(x) = 3x^2 + 5x + 7$. We substitute this into $g(x)$:

$g(f(x)) = g(3x^2 + 5x + 7)$

2. Now, substitute $3x^2 + 5x + 7$ wherever $x$ appears in $g(x) = 3x + 4$:

$g(3x^2 + 5x + 7) = 3(3x^2 + 5x + 7) + 4$

3. Distribute the constant:

$g(3x^2 + 5x + 7) = 9x^2 + 15x + 21 + 4$

4. Combine like terms:

$g(3x^2 + 5x + 7) = 9x^2 + 15x + 25$

Thus, the composition $(g \circ f)(x)$ is:

$(g \circ f)(x) = 9x^2 + 15x + 25$

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Final Answers:

(a) $(f \circ g)(x) = 27x^2 + 87x + 75$

(b) $(g \circ f)(x) = 9x^2 + 15x + 25$

Would you like any further clarification on the steps?

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Here are some related questions you might find useful:

1. How do we define function composition in general?
2. What happens when you compose two linear functions?
3. How does function composition affect the domain of the resulting function?
4. Can function composition be commutative, i.e., $f \circ g = g \circ f$?
5. How would you compute $(f \circ f)(x)$?

Tip: Always expand and simplify step-by-step in function composition to avoid calculation errors.