To graph the function [tex]\( h(x) \)[/tex] which is defined as:
[tex]\[
h(x) =
\begin{cases}
x & \text{if } x \neq 2 \\
3 & \text{if } x = 2
\end{cases}
\][/tex]
Let's break down the graphing process step-by-step:
1. Identify and graph the first case [tex]\( h(x) = x \)[/tex] when [tex]\( x \neq 2 \)[/tex]:
- This is a simple linear function. For any [tex]\( x \)[/tex] not equal to 2, the value of [tex]\( h(x) \)[/tex] is equal to [tex]\( x \)[/tex].
- Therefore, if you plot points for various [tex]\( x \)[/tex] values, such as [tex]\( -2, -1, 0, 1, 3, 4 \)[/tex], you get corresponding points on the line [tex]\( y = x \)[/tex]. For example:
- If [tex]\( x = -2 \)[/tex], then [tex]\( h(-2) = -2 \)[/tex]
- If [tex]\( x = 0 \)[/tex], then [tex]\( h(0) = 0 \)[/tex]
- If [tex]\( x = 3 \)[/tex], then [tex]\( h(3) = 3 \)[/tex]
- Plot these points and draw a straight line passing through them. However, be sure to leave a gap or an open circle at [tex]\( x = 2 \)[/tex], because the function is not defined as [tex]\( h(2) = 2 \)[/tex] here.
2. Identify and graph the second case [tex]\( h(x) = 3 \)[/tex] when [tex]\( x = 2 \)[/tex]:
- When [tex]\( x = 2 \)[/tex], the function [tex]\( h(x) \)[/tex] explicitly takes the value of 3.
- Therefore, at [tex]\( x = 2 \)[/tex], you plot a point at [tex]\( (2, 3) \)[/tex].
- Represent this point with a solid dot to indicate that the function is exactly 3 at [tex]\( x = 2 \)[/tex].
3. Combine both parts of the piecewise function into one graph:
- Draw the line [tex]\( y = x \)[/tex] with an open circle at [tex]\( (2, 2) \)[/tex].
- Represent the point [tex]\( (2, 3) \)[/tex] with a solid dot to indicate the specific value of the function at that point.
Visualizing the above steps on a coordinate plane:
- You will have a diagonally straight line [tex]\( y = x \)[/tex] passing through all points except at [tex]\( x = 2 \)[/tex] where there will be an open circle indicating the discontinuity.
- At the same x-value where the open circle is located (i.e., [tex]\( x = 2 \)[/tex]), place a solid dot at [tex]\( y = 3 \)[/tex].
The resulting graph will clearly show the piecewise nature of the function [tex]\( h(x) \)[/tex].
