Question

graph the following piecewise function.

$$ f ( x ) = left{ \begin{array} { l l } { 3 ( x + 4 ) + 1, } & { x geq - 4 } \\ { - 3 ( x + 4 ) + 1, } & { x < - 4 } end{array} ight. $$

Explanation:

Step1: Analyze the first part of the piece - wise function ($x\geq - 4$)

The function is \(y = 3(x + 4)+1=3x+12 + 1=3x + 13\). When \(x=-4\), \(y=3\times(-4)+13=1\). The slope \(m = 3\), so we can find another point. Let \(x = 0\), then \(y=3\times0+13 = 13\). Plot the line \(y = 3x + 13\) for \(x\geq - 4\), with a closed - circle at the point \((-4,1)\) (since \(x=-4\) is included in this part of the function).

Step2: Analyze the second part of the piece - wise function (\(x\lt - 4\))

The function is \(y=-3(x + 4)+1=-3x-12 + 1=-3x-11\). When \(x=-4\), \(y=-3\times(-4)-11=12 - 11 = 1\). The slope \(m=-3\). Let \(x=-5\), then \(y=-3\times(-5)-11=15 - 11 = 4\). Plot the line \(y=-3x - 11\) for \(x\lt - 4\), with an open - circle at the point \((-4,1)\) (since \(x=-4\) is not included in this part of the function).

Answer:

The graph consists of two lines: for \(x\geq - 4\), it is the line \(y = 3x + 13\) (closed - circle at \((-4,1)\)) and for \(x\lt - 4\), it is the line \(y=-3x - 11\) (open - circle at \((-4,1)\)).