Question

evaluate the compositions indicated for the given functions.

(f(x) = -4x^2), (g(x) = x^3 + 1)

((f \circ g)(2) = ?)

• (-4,095)
• (-324)
• (-255)
• (-196)

((g \circ f)(2) = ?)

• (-4,095)
• (-324)
• (-255)
• (-196)

Explanation:

⚡ Using what you learned: Evaluating Composite Functions

Step 1: Evaluate \( (f \circ g)(2) \)

\[
g(2) = 2^3 + 1 = 8 + 1 = 9
\]

\[
(f \circ g)(2) = f(g(2)) = f(9)
\]

\[
f(9) = -4(9)^2 = -4(81) = -324
\]

Step 2: Evaluate \( (g \circ f)(2) \)

\[
f(2) = -4(2)^2 = -4(4) = -16
\]

\[
(g \circ f)(2) = g(f(2)) = g(-16)
\]

\[
g(-16) = (-16)^3 + 1 = -4096 + 1 = -4095
\]

Answer:

• For \( (f \circ g)(2) \), the correct option is -324.
• For \( (g \circ f)(2) \), the correct option is -4,095.