- Define the three linear segments of the function: \(x+3\) for \(x<1\), \(-3x+7\) for \(1 \leq x \leq 5\), and \(x-7\) for \(x>5\).
- Recognize that the function is continuous at \(x=1\) because both segments evaluate to 4.
- Identify a jump discontinuity at \(x=5\), where the function value drops from -8 to -2.
- Graph the segments by plotting the rays and the connecting segment, using open circles for non-included endpoints at \(x=1\) and \(x=5\).
Explanation
- Analyzing the Piecewise Function Segments
The function \(f(x)\) is defined piecewise, meaning it follows different rules depending on the value of \(x\). We analyze these three segments separately to understand the graph's behavior:
- For \(x < 1\), the function is defined by \(f(x) = x + 3\). This is a line with a slope of 1. As \(x\) approaches 1 from the left, \(f(x)\) approaches \(1 + 3 = 4\). Because the interval is strictly less than 1 (\(x < 1\)), we use an open circle at \((1, 4)\) to indicate that this point is not included in this segment.
- For \(1 \leq x \leq 5\), the function is defined by \(f(x) = -3x + 7\). This is a line segment with a slope of -3. Evaluating at the endpoints, we get \(f(1) = -3(1) + 7 = 4\) and \(f(5) = -3(5) + 7 = -15 + 7 = -8\). We connect \((1, 4)\) and \((5, -8)\) with a solid line, using closed circles at both endpoints. Note that the value at \(x=1\) matches the limit from the previous segment, so the function is continuous there.
- For \(x > 5\), the function is defined by \(f(x) = x - 7\). This is a line with a slope of 1. As \(x\) approaches 5 from the right, \(f(x)\) approaches \(5 - 7 = -2\). Since the interval is strictly greater than 5 (\(x > 5\)), we use an open circle at \((5, -2)\) to indicate that this point is not included in this segment. This indicates a jump discontinuity at \(x=5\).
- Constructing the Graph
To construct the graph, follow these steps:
- Draw a line with a slope of 1, starting from the left and ending at the open circle at \((1, 4)\).
- Draw a solid line segment connecting the points \((1, 4)\) and \((5, -8)\). Ensure the point at \((1, 4)\) is filled in as a closed circle, which effectively connects the first two segments.
- Finally, place an open circle at \((5, -2)\) and draw a ray with a slope of 1 extending to the right.
This visualization clearly shows the continuity of the function at \(x=1\) and the jump discontinuity at \(x=5\).
Examples
Piecewise functions are used in real life to model situations that change their behavior at specific thresholds. For example, a shipping company might charge a flat rate for small packages, a different rate based on weight for medium packages, and a higher flat rate plus an extra charge for large, oversized packages. A graph of the shipping cost as a function of package weight would look exactly like a piecewise function, with different segments representing the different pricing rules and potential jumps at weight thresholds where the cost changes abruptly.