To graph the piecewise function \( f(x) \), let’s look at each part of the function separately and then combine them.
The function is given by:
[tex]\[ f(x) = \begin{cases}
x & \text{if } x < 2 \\
2 & \text{if } x \geq 2
\end{cases} \][/tex]
### Step-by-Step Graphing Instructions:
1. Graph the first part: \( f(x) = x \) for \( x < 2 \):
- This part is a linear function and would be a straight line with a slope of 1.
- To graph this, draw the line \( y = x \) starting from any \( x < 2 \).
2. Graph the value at the boundary \( x = 2 \):
- For \( x = 2 \), the function is defined as \( f(x) = 2 \).
- At \( x = 2 \), there will be a solid dot at the point (2, 2) because the function equals 2 for \( x \geq 2 \).
3. Graph the second part \( f(x) = 2 \) for \( x \geq 2 \):
- This represents a horizontal line at \( y = 2 \) starting from \( x = 2 \) and extending to infinity.
- Draw a horizontal line starting at the solid dot at (2, 2) and moving to the right.
### Summary of Steps on the Graph:
- Draw a straight line with a slope of 1 up to but not including \( x = 2 \).
- Place a solid dot at the point (2, 2).
- Draw a horizontal line starting at this point and extending to the right.
Now let’s compare your drawn graph with the given answer choices descriptions to find which one matches.
Without the actual text descriptions for graphs A, B, and C, we can't directly verify the correct answer here. However, based on the described steps:
- The correct graph will have a linear line y = x for \( x < 2 \).
- Starting precisely at \( x = 2 \), it will have a solid dot at (2, 2) and a horizontal line for \( x \geq 2 \).
You should look for the answer choice that correctly describes:
1. A sloped line matching \( y = x \) until \( x = 2 \).
2. A point where \( x = 2 \), \( y = 2 \) is clearly shown.
3. A horizontal line at \( y = 2 \) starting from \( x = 2 \).
Refer to the description that matches this graph !
