Short Answer 
The answer explains a piecewise function defined as f(x) = 3 for x < 1 and f(x) = x for x ‚a• 1. The graph features a constant line at y = 3 for x < 1 and transitions to a linear function beginning at (1, 1) for x ‚a• 1, with appropriate representation of open and closed circles indicating inclusivity of points.
Step 1: Understanding the Piecewise Function
The function given is a piecewise function represented as:
- f(x) = 3 for x < 1
- f(x) = x for x ‚a• 1
This means that for values of x less than 1, the function remains constant at 3, creating a horizontal line. For values of x greater than or equal to 1, the function follows the line f(x) = x.
Step 2: Analyzing the Graph on the Intervals
When we plot the function, we first look at the portion where x < 1. Here, the line remains at y = 3, and since it does not include 1, we represent it with an open circle at the point (1, 3). The graph continues left towards negative infinity.
For x ‚a• 1, we have a straight line starting from (1, 1) with a slope of 1. This indicates a filled circle at x = 1 because this value is included in the function definition.
Step 3: Final Conclusion about the Function Representation
In conclusion, the correct representation of the function is option A because:
- The horizontal line segment at y = 3 exists for values less than 1.
- The function transitions to a straight line through the point (1, 1) for values greater than or equal to 1.
Specifically, the hollow circle indicates the function does not include the point (1, 3), while the filled circle shows it does include the point (1, 1).