You provided two functions:

$f(x) = 2x + 8$ $g(x) = 4x - 8$

The questions ask for compositions of these functions:

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

This is the composition of $f$ and $g$, meaning we substitute $g(x)$ into $f(x)$.

$(f \circ g)(x) = f(g(x)) = f(4x - 8)$ Substitute $4x - 8$ into $f(x)$:

$f(4x - 8) = 2(4x - 8) + 8$ Simplify: $= 8x - 16 + 8 = 8x - 8$ So: $(f \circ g)(x) = 8x - 8$

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

This is the composition of $g$ and $f$, meaning we substitute $f(x)$ into $g(x)$.

$(g \circ f)(x) = g(f(x)) = g(2x + 8)$ Substitute $2x + 8$ into $g(x)$:

$g(2x + 8) = 4(2x + 8) - 8$ Simplify: $= 8x + 32 - 8 = 8x + 24$ So: $(g \circ f)(x) = 8x + 24$

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

This is a composition of $f$, $g$, and $f$, meaning we first find $f(x)$, then $g(f(x))$, and finally $f(g(f(x)))$.

From part (b), we already know that: $(g \circ f)(x) = 8x + 24$ Now substitute $8x + 24$ into $f(x)$:

$f(8x + 24) = 2(8x + 24) + 8$ Simplify: $= 16x + 48 + 8 = 16x + 56$ So: $(f \circ g \circ f)(x) = 16x + 56$

Summary of Results:

• $(f \circ g)(x) = 8x - 8$
• $(g \circ f)(x) = 8x + 24$
• $(f \circ g \circ f)(x) = 16x + 56$

Would you like more details on these compositions or have any questions?

Here are 5 follow-up questions for practice:

1. How would the results change if $f(x) = 3x + 5$ and $g(x) = x - 4$?
2. What is the inverse of $f(x) = 2x + 8$?
3. How would you graph $(f \circ g)(x)$?
4. What are the domains of $f(x)$, $g(x)$, and their compositions?
5. Can you find $(g \circ f \circ g)(x)$?

Tip: When solving function compositions, always start from the innermost function and work your way outward.