Question

suppose that the function f is defined as follows.

$$ f ( x ) = left{ \begin{array} { l l } { - 3 } & { \text { if } - 3.5 < x leq - 2.5 } \\ { - 2 } & { \text { if } - 2.5 < x leq - 1.5 } \\ { - 1 } & { \text { if } - 1.5 < x < - 0.5 } \\ { 0 } & { \text { if } - 0.5 leq x < 0.5 } \\ { 1 } & { \text { if } 0.5 leq x < 1.5 } end{array} ight. $$

graph the function f.

Explanation:

Step1: Analyze each piece of the function

For \( -3.5 < x\leq - 2.5\), \(y = - 3\). This is a horizontal line segment. At \(x=-3.5\), there is an open - circle (since \(x>-3.5\)) and at \(x = - 2.5\), there is a closed - circle (since \(x\leq - 2.5\)).

Step2: Analyze the second piece

For \( - 2.5 < x\leq - 1.5\), \(y=-2\). At \(x=-2.5\), there is an open - circle (since \(x > - 2.5\)) and at \(x=-1.5\), there is a closed - circle (since \(x\leq - 1.5\)).

Step3: Analyze the third piece

For \( - 1.5 < x < - 0.5\), \(y=-1\). At \(x=-1.5\) and \(x=-0.5\), there are open - circles (since \(x>-1.5\) and \(x < - 0.5\)).

Step4: Analyze the fourth piece

For \( - 0.5\leq x < 0.5\), \(y = 0\). At \(x=-0.5\), there is a closed - circle (since \(x\geq - 0.5\)) and at \(x = 0.5\), there is an open - circle (since \(x < 0.5\)).

Step5: Analyze the fifth piece

For \(0.5\leq x < 1.5\), \(y = 1\). At \(x = 0.5\), there is a closed - circle (since \(x\geq 0.5\)) and at \(x=1.5\), there is an open - circle (since \(x < 1.5\)).

To graph the function:

• Use a straight horizontal line for each interval.
• Place the appropriate open or closed circles at the endpoints of each interval according to the inequality signs in the piece - wise function definition.

Answer:

Graph the function \(y = f(x)\) with horizontal line segments:

• A horizontal line \(y=-3\) from \(x=-3.5\) (open - circle) to \(x=-2.5\) (closed - circle).
• A horizontal line \(y = - 2\) from \(x=-2.5\) (open - circle) to \(x=-1.5\) (closed - circle).
• A horizontal line \(y=-1\) from \(x=-1.5\) (open - circle) to \(x=-0.5\) (open - circle).
• A horizontal line \(y = 0\) from \(x=-0.5\) (closed - circle) to \(x=0.5\) (open - circle).
• A horizontal line \(y = 1\) from \(x=0.5\) (closed - circle) to \(x=1.5\) (open - circle).