Question

find the maximum and/or minimum and state whether it is a relative or absolute maximum/minimum. estimate if necessary. (8 points)
3)

sketch the graph of the piecewise function. (20 points)

1. $f(x)=\

$$\begin{cases}4x, & x < 0 \\\\ 2x + 1, & x > 1\\end{cases}$$

$

Explanation:

Step1: Analyze the first part of the piece - wise function ($x < 0$)

For the function \(y = 4x\) when \(x<0\).

• Choose some values of \(x\) less than \(0\). For example, if \(x=- 1\), then \(y = 4\times(-1)=-4\); if \(x = - 2\), then \(y=4\times(-2)=-8\).
• The function \(y = 4x\) is a straight - line with a slope \(m = 4\) and \(y\) - intercept \(b = 0\). Since \(x<0\), we draw the part of the line \(y = 4x\) to the left of the \(y\) - axis (excluding the origin because \(x

eq0\) in this part of the piece - wise function).

Step2: Analyze the second part of the piece - wise function ($x>1$)

For the function \(y=2x + 1\) when \(x>1\).

• Choose some values of \(x\) greater than \(1\). For example, if \(x = 2\), then \(y=2\times2+1=5\); if \(x=3\), then \(y=2\times3 + 1=7\).
• The function \(y=2x + 1\) is a straight - line with a slope \(m = 2\) and \(y\) - intercept \(b = 1\). Since \(x>1\), we draw the part of the line \(y=2x + 1\) to the right of \(x = 1\) (excluding the point \(x = 1\)).

Answer:

To sketch the graph:

1. For \(y = 4x,x<0\): Plot points like \((-1,-4),(-2,-8)\) and draw a straight line (dashed at \(x = 0\) since \(x

eq0\) for this part) to the left of the \(y\) - axis.

1. For \(y=2x + 1,x>1\): Plot points like \((2,5),(3,7)\) and draw a straight line (dashed at \(x = 1\) since \(x

eq1\) for this part) to the right of \(x = 1\).