We are given two functions, $f(x)$ and $g(x)$, and we are asked to find the value of $(f \circ g)(-4)$, which represents the composition of functions evaluated at $-4$.

Step 1: Understand the composition of functions

The notation $(f \circ g)(x)$ means $f(g(x))$, so we first need to evaluate $g(x)$ at $-4$ and then substitute that result into $f(x)$.

Step 2: Evaluate $g(x)$ at $-4$

We are given: $g(x) = 2x + 4$ Substitute $x = -4$: $g(-4) = 2(-4) + 4 = -8 + 4 = -4$

Step 3: Evaluate $f(x)$ at the result from Step 2

Now we substitute $g(-4) = -4$ into $f(x)$. The function $f(x)$ is given by: $f(x) = 3x^2 + 6x - 13$ Substitute $x = -4$: $f(-4) = 3(-4)^2 + 6(-4) - 13$ $f(-4) = 3(16) + (-24) - 13 = 48 - 24 - 13 = 11$

Final Answer:

$(f \circ g)(-4) = 11$

Would you like more details on any step or have questions about the method?

Here are five related questions to explore further:

1. How does function composition differ from simple addition or multiplication of functions?
2. What are the domain and range considerations for composed functions like $(f \circ g)(x)$?
3. How can we decompose a composed function into simpler functions?
4. What is the geometric interpretation of composing functions?
5. Can you solve for $(g \circ f)(x)$ instead, and how does it differ?

Tip: Always evaluate the inner function first when dealing with composition to avoid confusion.