We are given the functions $f(x) = x + 2$ and $g(x) = 4x + 4$. We are tasked with finding four composite functions and their values.

Let's go through each part step by step:

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

This represents $f(g(x))$, meaning we substitute $g(x)$ into $f(x)$.

Given:

• $f(x) = x + 2$
• $g(x) = 4x + 4$

Now substitute $g(x)$ into $f(x)$: $f(g(x)) = f(4x + 4) = (4x + 4) + 2 = 4x + 6$ So, $(f \circ g)(x) = 4x + 6$.

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

This represents $g(f(x))$, meaning we substitute $f(x)$ into $g(x)$.

Given:

• $f(x) = x + 2$
• $g(x) = 4x + 4$

Now substitute $f(x)$ into $g(x)$: $g(f(x)) = g(x + 2) = 4(x + 2) + 4 = 4x + 8 + 4 = 4x + 12$ So, $(g \circ f)(x) = 4x + 12$.

c. $(f \circ g)(0)$

This represents $f(g(0))$.

First, find $g(0)$: $g(0) = 4(0) + 4 = 4$

Now substitute this into $f(x)$: $f(g(0)) = f(4) = 4 + 2 = 6$ So, $(f \circ g)(0) = 6$.

d. $(g \circ f)(0)$

This represents $g(f(0))$.

First, find $f(0)$: $f(0) = 0 + 2 = 2$

Now substitute this into $g(x)$: $g(f(0)) = g(2) = 4(2) + 4 = 8 + 4 = 12$ So, $(g \circ f)(0) = 12$.

Final Answers:

• a. $(f \circ g)(x) = 4x + 6$
• b. $(g \circ f)(x) = 4x + 12$
• c. $(f \circ g)(0) = 6$
• d. $(g \circ f)(0) = 12$

Would you like more details or have any questions?

Here are 5 related questions for further practice:

1. What is $(f \circ f)(x)$ for $f(x) = x + 2$?
2. What is $(g \circ g)(x)$ for $g(x) = 4x + 4$?
3. How do you find the inverse of the function $f(x) = x + 2$?
4. How does changing the slope of $g(x)$ affect $(f \circ g)(x)$?
5. If $f(x)$ were $x^2 + 2$, how would that change $(f \circ g)(x)$?

Tip: Always simplify inside the composite function first before applying the outer function.