Let $g ( x ) = \left\{ \begin{array} { l l } { 2 } & { \text { for } } & { 0 \leq x \leq 3 } \\ { 3 } & { \text { for } } & { 3 < x \leq 5 } \\ { 7 } & { \text { for } } & { 5 < x \leq 8 } \end{array} \right.$

1. Sketch the graph of $g(x)$. Is this function continuous? Explain.

   Hint (a):

   Continuous functions must have connected pieces. There can be no jumps or holes. What happens to $g(x)$ at $x = 3$ and at $x=5$?

2. Shade the area between $g(x)$ and the $x$-axis. Find $A(g, 0 ≤ x ≤ 8)$.

   Hint (b):

   Area of Rectangle A: $2 × 3 = 6$ units^{$^2$}
   Area of Rectangle B: $3 × 2 = 6$ units^{$^2$}
   Area of Rectangle C: $7 × 3 = 21$ units^{$^2$}

   Answer (b):

   Rectangle A + Rectangle B + Rectangle C $= 33$ units^{$^2$}

3. The equation $g(x)$ is an example of a step function. Why do you think it is called a step function?

   Hint (c):

   The shape of the graph should help answer this question.