- The function is defined by two linear segments: \(f(x) = 2x + 1\) for \(x < 1\) and \(f(x) = -2x + 1\) for \(x \geq 1\).
- For \(x < 1\), the graph is a line segment with a slope of 2 passing through \((0, 1)\), ending in an open circle at \((1, 3)\).
- For \(x \geq 1\), the graph is a ray starting with a closed circle at \((1, -1)\) with a slope of -2.
- The graph shows the discontinuity at \(x=1\), visually representing how the output jumps between the two linear behaviors.
Explanation
- Analyzing the Piecewise Function
The function \(f(x)\) is a piecewise-defined function. It behaves differently depending on the value of \(x\).
The function is given by:
\[f(x) = \begin{cases} 2x + 1 & \text{for } x < 1 \ -2x + 1 & \text{for } x \geq 1 \end{cases}\]
This means for all real numbers \(x\) strictly less than 1, we use the linear equation \(y = 2x + 1\). For all real numbers \(x\) greater than or equal to 1, we use the linear equation \(y = -2x + 1\). The point \(x = 1\) is the boundary where the rule changes.
- Calculating Key Points for the Graph
To graph the function, we examine each part:
- For \(x < 1\), the expression is \(y = 2x + 1\).
- At \(x = 0\), \(y = 2(0) + 1 = 1\). So, the point \((0, 1)\) is on the graph.
- As \(x\) approaches 1 from the left, \(y\) approaches \(2(1) + 1 = 3\). Since the inequality is strictly \(x < 1\), this point \((1, 3)\) will be an open circle.
- For \(x \geq 1\), the expression is \(y = -2x + 1\).
- At \(x = 1\), \(y = -2(1) + 1 = -1\). Since the inequality is \(x \geq 1\), this point \((1, -1)\) will be a closed circle.
- At \(x = 2\), \(y = -2(2) + 1 = -3\). So, the point \((2, -3)\) is on the graph.
- Constructing the Graph
To represent this function on a coordinate plane:
- Plot the line \(y = 2x + 1\) for values of \(x\) to the left of 1. Start from the left and draw the segment up to \((1, 3)\), placing an open circle at \((1, 3)\) because \(x < 1\).
- Plot the line \(y = -2x + 1\) for values of \(x\) greater than or equal to 1. Start with a closed circle at \((1, -1)\) and draw the ray moving down and to the right through points like \((2, -3)\) and \((3, -5)\).
The graph consists of two distinct linear parts separated by a gap at \(x = 1\).
Examples
Piecewise functions are commonly used in modeling real-world scenarios where costs or rates change at certain thresholds. For example, consider a shipping company that charges a base fee plus a rate based on the weight of a package. The cost function might be defined by one linear rate for packages under a certain weight threshold and a different, potentially higher or lower, rate for heavier packages. Graphing such a function allows businesses and customers to visualize how the total shipping cost changes as the weight increases, helping to identify the exact cost at any given weight.