Question: Given the functions $f(x) = 3x + 2$ and $g(x) = x^2 - 5$, determine the value of the following composition of the functions: $g(f(x))$
1. **State the problem:** We need to find the composition $g(f(x))$, which means we substitute $f(x)$ into $g(x)$.
2. **Recall the functions:**
$f(x) = 3x + 2$
$g(x) = x^2 - 5$
3. **Write the composition formula:**
$$g(f(x)) = g(3x + 2) = (3x + 2)^2 - 5$$
4. **Expand the square:**
$$(3x + 2)^2 = (3x)^2 + 2 \cdot 3x \cdot 2 + 2^2 = 9x^2 + 12x + 4$$
5. **Substitute back:**
$$g(f(x)) = 9x^2 + 12x + 4 - 5 = 9x^2 + 12x - 1$$
6. **Final answer:**
$$g(f(x)) = 9x^2 + 12x - 1$$
This matches the answer choice: **9x^2 + 12x - 1**.