We are asked to find the value of $(f \circ g)(5)$, which represents the composition of the functions $f(x)$ and $g(x)$.

Given functions:

• $f(x) = 3x^2 + 6x + 15$
• $g(x) = -x + 1$

Composition process:

1. First, we calculate $g(5)$, because the composition $(f \circ g)(5)$ means we apply $g(x)$ first: $g(5) = -(5) + 1 = -5 + 1 = -4$

2. Next, we substitute $g(5) = -4$ into $f(x)$ to find $f(g(5))$, which is $f(-4)$: $f(-4) = 3(-4)^2 + 6(-4) + 15$ $f(-4) = 3(16) + 6(-4) + 15 = 48 - 24 + 15 = 39$

Final result:

$(f \circ g)(5) = 39$

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

Here are 5 related questions to consider:

1. How do you compute the composition of two functions in general?
2. What happens when you compose functions in the opposite order, $(g \circ f)(x)$?
3. How does the function composition affect the domain of the composed function?
4. What is the geometric interpretation of composing two functions?
5. How would you find the derivative of a composed function $(f \circ g)(x)$?

Tip: Always apply the inner function first when dealing with function composition, and substitute its result into the outer function.