Question

let f and g be the functions in the table below.

x	f(x)	f(x)	g(x)	g(x)
2	1	5	3	7
3	2	7	1	9

(a) if f(x)=f(f(x)), find f(3).
f(3)=
(b) if g(x)=g(g(x)), find g(2).
g(2)=

Explanation:

Step1: Recall the chain - rule

The chain - rule states that if $y = f(u)$ and $u = g(x)$, then $\frac{dy}{dx}=f^{\prime}(u)g^{\prime}(x)$. For $F(x)=f(f(x))$, let $u = f(x)$, so $F^{\prime}(x)=f^{\prime}(f(x))\cdot f^{\prime}(x)$.

Step2: Find $F^{\prime}(3)$

First, when $x = 3$, we need to find $f(3)$. From the table, $f(3)=2$. Then $F^{\prime}(3)=f^{\prime}(f(3))\cdot f^{\prime}(3)$. Since $f(3) = 2$ and $f^{\prime}(3)=7$, and $f^{\prime}(2)=5$, we have $F^{\prime}(3)=f^{\prime}(2)\cdot f^{\prime}(3)=5\times7 = 35$.

Step3: Recall the chain - rule for $G(x)$

For $G(x)=g(g(x))$, let $u = g(x)$, so $G^{\prime}(x)=g^{\prime}(g(x))\cdot g^{\prime}(x)$.

Step4: Find $G^{\prime}(2)$

When $x = 2$, we first find $g(2)$. From the table, $g(2)=3$. Then $G^{\prime}(2)=g^{\prime}(g(2))\cdot g^{\prime}(2)$. Since $g(2)=3$ and $g^{\prime}(2)=7$, and $g^{\prime}(3)=9$, we have $G^{\prime}(2)=g^{\prime}(3)\cdot g^{\prime}(2)=9\times7 = 63$.

Answer:

(a) $F^{\prime}(3)=35$
(b) $G^{\prime}(2)=63$