To graph the given piecewise function:

Function Definition:

1. $f(x) = -2x$ if $x < 1$
2. $f(x) = 0.5x + 1$ if $x \geq 1$

Step-by-step Procedure:

1. For $x < 1$:

   • This part of the function is $f(x) = -2x$, a straight line with slope $-2$ and passing through the origin.
   • Since $x < 1$, only plot this line for values of $x$ less than 1.
   • For $x = 1$, $f(x) = -2(1) = -2$ (not included, represented by an open circle at $(1, -2)$).

2. For $x \geq 1$:

   • This part of the function is $f(x) = 0.5x + 1$, a straight line with slope $0.5$ and $y$-intercept at $(0, 1)$.
   • Since $x \geq 1$, only plot this line for values of $x \geq 1$.
   • For $x = 1$, $f(x) = 0.5(1) + 1 = 1.5$ (included, represented by a closed circle at $(1, 1.5)$).

Plot Points:

• For $f(x) = -2x$ (when $x < 1$):

  • At $x = 0$, $f(x) = 0$.
  • At $x = -1$, $f(x) = -2(-1) = 2$.

• For $f(x) = 0.5x + 1$ (when $x \geq 1$):

  • At $x = 1$, $f(x) = 1.5$.
  • At $x = 2$, $f(x) = 0.5(2) + 1 = 2$.
  • At $x = 3$, $f(x) = 0.5(3) + 1 = 2.5$.

Notes for Graphing:

1. Mark an open circle at $(1, -2)$ for $f(x) = -2x$.
2. Mark a closed circle at $(1, 1.5)$ for $f(x) = 0.5x + 1$.
3. Draw the respective lines in their domains.

Would you like a graphical representation or further clarification?

Here are some questions to expand on this concept:

1. How can we determine if a circle at a boundary is open or closed?
2. What does the slope of a line indicate about its behavior?
3. How would the graph change if the conditions for $x$ were reversed?
4. How do we evaluate the continuity of a piecewise function at the boundaries?
5. What are the implications of having different slopes in the two parts of the function?

Tip: Always test the boundary point in both pieces to ensure you correctly represent its inclusion or