Question

1. graph the piecewise function given by

$$ g ( x ) = left\\{ \begin{array} { l l } { 4 } & { - 9 leq x leq - 5 } \\ { x } & { - 5 < x leq 2 } \\ { - 2 } & { x > 2 } end{array} ight. $$

Explanation:

Step1: Graph \( y = 4\) for \(-9\leq x\leq - 5\)

For the interval \(-9\leq x\leq -5\), the function \(g(x)=4\). This is a horizontal line segment. Plot the points \((-9,4)\) and \((-5,4)\) and connect them with a straight - line segment.

Step2: Graph \(y = x\) for \(-5\lt x\leq2\)

For the interval \(-5\lt x\leq2\), the function \(g(x)=x\). When \(x=-5\), \(y = - 5\) (but since \(x=-5\) is not included in this sub - function, we have an open - circle at \((-5,-5)\)). When \(x = 2\), \(y=2\). Plot the points \((-4,-4)\), \((-3,-3)\), \((-2,-2)\), \((-1,-1)\), \((0,0)\), \((1,1)\), \((2,2)\) and draw a straight line through these points (with an open - circle at \(x=-5\)).

Step3: Graph \(y=-2\) for \(x > 2\)

For the interval \(x>2\), the function \(g(x)=-2\). This is a horizontal line. Plot a point at \((3,-2)\) and draw a straight - line to the right with an open - circle at \(x = 2\) (since \(x = 2\) is not included in this sub - function).

Answer:

The graph of the piece - wise function \(g(x)\) consists of a horizontal line segment \(y = 4\) from \(x=-9\) to \(x=-5\), a line \(y=x\) from \(x=-5\) (open - circle) to \(x = 2\), and a horizontal line \(y=-2\) for \(x>2\) (open - circle at \(x = 2\)).