This worksheet provides a comprehensive exploration of piecewise functions, including definitions, graphing techniques, and practical applications. Whether you're a student looking to solidify your understanding or a teacher seeking engaging materials, this resource offers a structured approach to mastering this essential mathematical concept. We'll cover everything from basic evaluation to more complex problem-solving, complete with detailed answers to help you check your work.
What are Piecewise Functions?
A piecewise function is a function defined by multiple subfunctions, each applicable over a specified interval of the domain. These intervals are non-overlapping and together cover the entire domain of the function. Think of it as a function assembled from different pieces, each with its own rule. The notation often involves using curly braces {} to list the different subfunctions and their corresponding domains.
Example:
A simple piecewise function might be defined as:
f(x) = {
x + 2, if x < 0
x² , if x ≥ 0
}
This means that for values of x less than 0, the function follows the rule f(x) = x + 2. For values of x greater than or equal to 0, the rule changes to f(x) = x².
Evaluating Piecewise Functions
Evaluating a piecewise function involves determining which subfunction applies to the given input value and then applying that subfunction's rule.
Example:
Using the function defined above:
f(-2): Since -2 < 0, we use the first subfunction:f(-2) = -2 + 2 = 0f(0): Since 0 ≥ 0, we use the second subfunction:f(0) = 0² = 0f(3): Since 3 ≥ 0, we use the second subfunction:f(3) = 3² = 9
Graphing Piecewise Functions
Graphing piecewise functions requires graphing each subfunction within its designated interval. Pay close attention to the endpoints of each interval, as they often determine whether the point is included (closed circle) or excluded (open circle) on the graph.
Example:
Let's graph the function:
f(x) = {
2x + 1, if x < 1
x - 1, if x ≥ 1
}
-
Graph 2x + 1 for x < 1: This is a line with a slope of 2 and a y-intercept of 1. However, we only graph it for values of x less than 1. At x = 1, we'll have an open circle because the inequality is strict.
-
Graph x - 1 for x ≥ 1: This is a line with a slope of 1 and a y-intercept of -1. We graph this line starting at x = 1 and extending to the right. At x=1, we use a closed circle because the inequality includes the equals sign.
By combining these two graphs, you create the complete graph of the piecewise function.
Worksheet Problems (with Answers)
Problem 1: Evaluate the function g(x) = { 3x - 1, if x ≤ 2; x² + 1, if x > 2} at the following points:
a) g(0) Answer: -1 b) g(2) Answer: 5 c) g(4) Answer: 17
Problem 2: Graph the function h(x) = { |x|, if x < 1; 2, if 1 ≤ x ≤ 3; x -1, if x > 3}
(Answer: This requires a graphical representation. You should see a V-shaped graph for |x| up to x=1 (open circle at x=1), a horizontal line at y=2 from x=1 (closed circle) to x=3 (closed circle), and a line with a slope of 1 and y-intercept of -1 starting from x=3 (open circle) extending to the right.)
Problem 3: Write a piecewise function that represents the following scenario: A taxi charges $3 for the first mile and $2 for each additional mile.
(Answer: Let C(m) be the cost for m miles. Then C(m) = { 3, if 0 < m ≤ 1; 3 + 2(m-1), if m > 1 })
This worksheet provides a starting point for understanding piecewise functions. Further exploration may include more complex scenarios involving absolute value functions, step functions, and applications in real-world problems. Remember to practice regularly to build proficiency!
