Question

the following table lists the values of functions f and g, and of their derivatives, f and g, for the x-values -2 and 4.

x	f(x)	g(x)	f(x)	g(x)
4	-4	-2	0	8

let function f be defined as f(x) = f(g(x)).

f(4) = \boxed{}

Explanation:

Step1: Apply Chain Rule

To find \( F'(x) \) where \( F(x) = f(g(x)) \), we use the chain rule: \( F'(x) = f'(g(x)) \cdot g'(x) \).

Step2: Evaluate at \( x = 4 \)

First, find \( g(4) \) from the table. When \( x = 4 \), \( g(4) = -2 \).

Then, find \( f'(g(4)) = f'(-2) \). From the table, when \( x = -2 \), \( f'(-2) = 1 \).

Next, find \( g'(4) \) from the table. When \( x = 4 \), \( g'(4) = 8 \).

Step3: Multiply the results

Using the chain rule formula, \( F'(4) = f'(g(4)) \cdot g'(4) = f'(-2) \cdot g'(4) = 1 \cdot 8 = 8 \).

Answer:

\( 8 \)