Question

Graph function
Z(x)=
-3 for -3<x<-2 ?
X for -2<=x<1 ?
-x^2+1 for x>=1 ?

Answer 1

To graph the piecewise function \( Z(x) \), we will break it down according to its definition over the specified intervals:

1. **For \(-3 < x < -2\)**:
\[
Z(x) = -3
\]
This segment is a horizontal line at \( Z = -3 \) from \( x = -3 \) to \( x = -2 \). It does not include the endpoints.

2. **For \(-2 \leq x < 1\)**:
\[
Z(x) = x
\]
This segment is a straight line with a slope of 1, starting from the point \((-2, -2)\) (inclusive) and increasing to just below \( (1, 1) \) (not including \( 1 \)).

3. **For \(x \geq 1\)**:
\[
Z(x) = -x^2 + 1
\]
This is a downward-opening parabola. The vertex of the parabola occurs at \( (0, 1) \), but since we only consider values for \( x \) starting at \( 1 \), we can compute this segment specifically:
- At \( x=1 \):
\[
Z(1) = -1^2 + 1 = 0
\]
- At \( x=2 \):
\[
Z(2) = -2^2 + 1 = -3
\]
- At \( x=3 \):
\[
Z(3) = -3^2 + 1 = -8
\]
The parabola will continue to decrease as \( x \) increases.

### Graphing the Function

1. **Plot the horizontal line** segment from \( (-3, -3) \) to \( (-2, -3) \).
2. **Plot the line segment** from \( (-2, -2) \) to \( (1, 1) \), including the point at \( (-2, -2) \) but not at \( (1, 1) \).
3. **Plot the downward parabola** starting at \( (1, 0) \). Draw it curving downwards and shading in the area below it.

### Summary of Key Points
- Two open intervals: \( (-3, -2) \) and \( (1, \infty) \)
- At \(-2\), \( Z(-2) = -2 \) (inclusive)
- At \( x = 1, Z(1) = 0 \) (inclusive)

You can create the graph using graphing software or plot it manually on graph paper following the guidelines above.