Question

graph of f
the graph of the function f, consisting of two line segments, is shown in the figure above. let g be the function given by ( g ( x ) = 2 x + 1 ), and let h be the function given by ( h ( x ) = f ( g ( x ) ) ). what is the value of ( h ^ { prime } ( 1 ) )?
a -4
b -2
c 4
d 6

Explanation:

Step1: Apply the chain rule

The chain rule states that \(h^{\prime}(x)=f^{\prime}(g(x))\cdot g^{\prime}(x)\).
First, find \(g^{\prime}(x)\). Since \(g(x) = 2x+1\), then \(g^{\prime}(x)=2\).

Step2: Find \(g(1)\)

Substitute \(x = 1\) into \(g(x)\): \(g(1)=2\times1 + 1=3\).

Step3: Find \(f^{\prime}(3)\)

The function \(f(x)\) has two line - segments.
For \(x\in[0,1]\), the slope \(m_1=\frac{4 - 1}{1-0}=3\).
For \(x\in[1,4]\), the slope \(m_2=\frac{-2 - 4}{4 - 1}=\frac{-6}{3}=-2\).
Since \(g(1) = 3\) and \(3\in[1,4]\), \(f^{\prime}(3)=-2\).

Step4: Calculate \(h^{\prime}(1)\)

Substitute into the chain - rule formula: \(h^{\prime}(1)=f^{\prime}(g(1))\cdot g^{\prime}(1)\).
We know that \(g^{\prime}(1) = 2\) and \(f^{\prime}(g(1))=f^{\prime}(3)=-2\).
So \(h^{\prime}(1)=(-2)\times2=-4\).

Answer:

A. - 4