Question

1. the graph of the function ( g ) is shown above. if ( f ) is the function given by ( f(x)=g(g(x)) ), what is the value of ( f^{prime}(0) )?

Explanation:

Step1: Find \( g(0) \)

From the graph of \( g \), when \( x = 0 \), the \( y \)-value (which is \( g(0) \)) is 3. So \( g(0)=3 \).

Step2: Find the derivative of \( f(x) \) using the chain rule

The function \( f(x)=g(g(x)) \). By the chain rule, \( f'(x)=g'(g(x))\cdot g'(x) \).

Step3: Evaluate \( f'(0) \)

Substitute \( x = 0 \) into the derivative: \( f'(0)=g'(g(0))\cdot g'(0) \). We know \( g(0) = 3 \), so this becomes \( f'(0)=g'(3)\cdot g'(0) \).

Step4: Find \( g'(0) \)

To find \( g'(0) \), we look at the slope of the graph of \( g \) at \( x = 0 \). The graph of \( g \) near \( x = 0 \) is a line segment from \( (-2, -1) \) to \( (1, 5) \). The slope \( m \) between two points \( (x_1,y_1) \) and \( (x_2,y_2) \) is \( \frac{y_2 - y_1}{x_2 - x_1} \). Using \( (-2, -1) \) and \( (1, 5) \), the slope is \( \frac{5 - (-1)}{1 - (-2)}=\frac{6}{3}=2 \). So \( g'(0)=2 \).

Step5: Find \( g'(3) \)

To find \( g'(3) \), we look at the slope of the graph of \( g \) at \( x = 3 \). The graph of \( g \) for \( x \geq 1 \) is a line segment from \( (1, 5) \) to \( (6, 0) \). The slope between \( (1, 5) \) and \( (6, 0) \) is \( \frac{0 - 5}{6 - 1}=\frac{-5}{5}=-1 \). So \( g'(3)=-1 \) (since \( 3 \) is in the interval \( [1, 6] \)).

Step6: Calculate \( f'(0) \)

Substitute \( g'(3)=-1 \) and \( g'(0)=2 \) into \( f'(0)=g'(3)\cdot g'(0) \): \( f'(0)=(-1)\cdot 2=-2 \).

Answer:

\(-2\)