To graph the piecewise function [tex]\( f(x) = \left\{\begin{array}{l}3x \text{ if } x < 3 \\ 2x \text{ if } x \geq 3\end{array}\right. \)[/tex], let's follow these steps:
1. Understand the Function:
- For [tex]\( x < 3 \)[/tex], the function is defined as [tex]\( f(x) = 3x \)[/tex].
- For [tex]\( x \geq 3 \)[/tex], the function is defined as [tex]\( f(x) = 2x \)[/tex].
2. Graph [tex]\( f(x) = 3x \)[/tex] for [tex]\( x < 3 \)[/tex]:
- This is a linear function with a slope of 3.
- On the interval [tex]\( (-\infty, 3) \)[/tex], plot the line [tex]\( y = 3x \)[/tex].
To plot it, consider a couple of points:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 3 \times 0 = 0 \)[/tex]. So, the point [tex]\( (0, 0) \)[/tex] is on this part of the graph.
- When [tex]\( x = 2 \)[/tex], [tex]\( y = 3 \times 2 = 6 \)[/tex]. So, the point [tex]\( (2, 6) \)[/tex] is on this part of the graph.
Connect these points with a line but stop at [tex]\( x = 3 \)[/tex] and do not include it (an open circle at [tex]\( x = 3 \)[/tex]).
3. Graph [tex]\( f(x) = 2x \)[/tex] for [tex]\( x \geq 3 \)[/tex]:
- This is a linear function with a slope of 2.
- On the interval [tex]\( [3, \infty) \)[/tex], plot the line [tex]\( y = 2x \)[/tex].
To plot it, consider a couple of points:
- When [tex]\( x = 3 \)[/tex], [tex]\( y = 2 \times 3 = 6 \)[/tex]. So, the point [tex]\( (3, 6) \)[/tex] is on this part of the graph.
- When [tex]\( x = 4 \)[/tex], [tex]\( y = 2 \times 4 = 8 \)[/tex]. So, the point [tex]\( (4, 8) \)[/tex] is on this part of the graph.
Connect these points with a line starting at [tex]\( x = 3 \)[/tex] and include the point [tex]\( (3, 6) \)[/tex] (a closed circle at [tex]\( x = 3 \)[/tex]).
4. Summary of the Graph:
- From [tex]\( x = -\infty \)[/tex] to [tex]\( x = 3 \)[/tex], draw the line [tex]\( y = 3x \)[/tex] but end with an open circle at [tex]\( (3, 9) \)[/tex].
- From [tex]\( x = 3 \)[/tex] to [tex]\( x = \infty \)[/tex], draw the line [tex]\( y = 2x \)[/tex] beginning with a closed circle at [tex]\( (3, 6) \)[/tex].
When you graph this on a piece of paper, you will see:
- A line starting from [tex]\( (-\infty, 0) \)[/tex] to [tex]\( (3, 9) \)[/tex] with an open circle at [tex]\( (3, 9) \)[/tex].
- Another line starts from [tex]\( (3, 6) \)[/tex] with a closed circle and extends to [tex]\( (\infty, \infty) \)[/tex].
To choose which answer choice matches, simply look for the piecewise plot that has these characteristics.
