We are tasked with finding $(f \circ g)(5)$. This represents the composition of two functions $f(x)$ and $g(x)$, which means we first evaluate $g(5)$ and then substitute that result into $f(x)$.

Given:

• $f(x) = x^2 + 7x + 4$
• $g(x) = -4x + 11$

Step 1: Find $g(5)$

Substitute $x = 5$ into $g(x)$: $g(5) = -4(5) + 11 = -20 + 11 = -9$

Step 2: Find $f(g(5)) = f(-9)$

Now substitute $x = -9$ into $f(x)$: $f(-9) = (-9)^2 + 7(-9) + 4$ First, calculate the terms: $(-9)^2 = 81$ $7(-9) = -63$ $f(-9) = 81 - 63 + 4 = 22$

Final Answer:

$(f \circ g)(5) = 22$

Would you like further details or have any questions?

Here are 5 related questions:

1. How do we interpret function composition?
2. What happens when you reverse the order of composition, $(g \circ f)(x)$?
3. Can function composition be applied to non-linear functions?
4. What is the significance of function domains in composition?
5. How do you graph the composition of two functions?

Tip: When composing functions, always apply the inner function first and use its result for the outer function!