Question

Question content area left
Part 1
Graph the​ piecewise-defined function.
f left parenthesis x right parenthesis equals
left brace Start 2 By 3 Matrix 1st Row 1st Column x plus 6 2nd Column if 3rd Column x less than minus 1 2nd Row 1st Column negative 2 x plus 4 2nd Column if 3rd Column x greater than or equals minus 1 EndMatrix

Answer 1

To graph the piecewise function \( f(x) \), we'll consider the function's definition in two parts based on the value of \( x \):

1. **For \( x < -1 \)**:
\[
f(x) = x + 6
\]
This is a linear function with a slope of 1 and a y-intercept of 6.

2. **For \( x \geq -1 \)**:
\[
f(x) = -2x + 4
\]
This is also a linear function with a slope of -2 and a y-intercept of 4.

### Steps to Graph the Function

1. **Graph the first part** \( f(x) = x + 6 \) for \( x < -1 \):
- Start by finding two points.
- At \( x = -2 \):
\[
f(-2) = -2 + 6 = 4 \quad \text{(point: } (-2, 4)\text{)}
\]
- At \( x = -1 \):
\[
f(-1) = -1 + 6 = 5 \quad \text{(this point is not included as } x \text{ must be less than } -1\text{)}
\]

- Plot the point \((-2, 4)\) and draw a line extending leftward from this point (but leave an open circle at \((-1, 5)\) since \( f(-1) \) is not defined as \( x \) cannot equal \(-1\) in this part).

2. **Graph the second part** \( f(x) = -2x + 4 \) for \( x \geq -1 \):
- Find two points.
- At \( x = -1 \):
\[
f(-1) = -2(-1) + 4 = 2 + 4 = 6 \quad \text{(point: } (-1, 6)\text{)}
\]
- At \( x = 0 \):
\[
f(0) = -2(0) + 4 = 4 \quad \text{(point: } (0, 4)\text{)}
\]

- Plot the point \((-1, 6)\) with a closed circle (since this point is included) and the point \((0, 4)\). Then draw a line extending rightward from \((-1, 6)\).

### Final Graph

- The graph will have:
- A line segment with a slope of 1 starting from point \((-2, 4)\) and extending leftwards, stopping at \((-1, 5)\) (open circle).
- A line with slope -2 starting from \((-1, 6)\) (closed circle) and extending rightwards through point \((0, 4)\).

Make sure to label your axes and mark the specific points mentioned for clarity. The final graph will visually represent the piecewise function as described.