Question

the following table lists the values of functions f and h, and of their derivatives, f and h, for the x-values -1 and 2.

x	f(x)	h(x)	f(x)	h(x)
2	3	-1	1	6

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

g(2) = \boxed{\space}

Explanation:

Step1: Apply Chain Rule

To find \( G'(x) \) where \( G(x) = f(h(x)) \), we use the chain rule. The chain rule states that \( G'(x) = f'(h(x)) \cdot h'(x) \).

Step2: Evaluate at \( x = 2 \)

First, find \( h(2) \) from the table. When \( x = 2 \), \( h(2) = -1 \).

Step3: Find \( f'(h(2)) \)

Now, \( h(2) = -1 \), so we need \( f'(-1) \). From the table, when \( x = -1 \), \( f'(-1) = -5 \).

Step4: Find \( h'(2) \)

From the table, when \( x = 2 \), \( h'(2) = 6 \).

Step5: Multiply the results

Using the chain rule formula \( G'(2) = f'(h(2)) \cdot h'(2) \), substitute the values we found: \( G'(2) = f'(-1) \cdot h'(2) = -5 \cdot 6 \).

Step6: Calculate the product

\( -5 \cdot 6 = -30 \).

Answer:

\(-30\)