Question

graph y\ =\ \left\{30x\ if\ 0\ <\ x\ \le\ 3\ 75\ if\ 3\ <\ x\ \le\ 7\ 20x\ if\ x\ >\ 7\right\}

Answer 1

To graph the piecewise function defined by

\[
y =
\begin{cases}
30x & \text{if } 0 < x \leq 3 \\
75 & \text{if } 3 < x \leq 7 \\
20x & \text{if } x > 7
\end{cases}
\]

we'll analyze each piece of the function separately and then combine them to create a complete graph.

### 1. For \( 0 < x \leq 3 \): \( y = 30x \)

- This is a linear function with a slope of 30.
- The point at \( x = 0 \) is not included (open interval), and at \( x = 3 \):
\[
y = 30(3) = 90
\]
- So we have the line segment from just above the origin to the point \( (3, 90) \).

### 2. For \( 3 < x \leq 7 \): \( y = 75 \)

- This is a constant function.
- The value of \( y \) is 75 from slightly above \( x = 3 \) to \( x = 7 \).
- At \( x = 3 \), \( y \) will not include the point because the interval is open at this endpoint.
- At \( x = 7 \):
\[
y = 75
\]
- So we have a horizontal line segment from just above \( (3, 75) \) to \( (7, 75) \) inclusive.

### 3. For \( x > 7 \): \( y = 20x \)

- This is again a linear function, with a slope of 20.
- At \( x = 7 \), the function transitions as follows:
\[
y = 20(7) = 140
\]
- So at this point, the function starts from \( (7, 140) \) and heads off to the right.

### Final Points for Graphing:

- **For \( 0 < x \leq 3 \)**: Draw a line segment from \( (0, 0) \) (open circle) to \( (3, 90) \) (closed circle).
- **For \( 3 < x \leq 7 \)**: Draw a horizontal line segment from just above \( (3, 75) \) (open circle) to \( (7, 75) \) (closed circle).
- **For \( x > 7 \)**: Start from the point \( (7, 140) \) (closed circle) and draw the line \( y = 20x \) extending to the right.

### Graph:

Here's a rough representation of how the graph would look like:

1. For the first segment, from \( x = 0 \) to \( x = 3 \):
- Starts at an open circle at (0,0) and ends at a closed circle at (3,90).
2. From \( x = 3 \) to \( x = 7 \):
- A horizontal line across \( y = 75 \) from just above \( (3, 75) \) to \( (7, 75) \).
3. For \( x > 7 \):
- Begins at (7, 140) with a closed circle and continues upwards with slope of 20.

To visualize, you can sketch based on the information provided.