piecewise functions worksheet and answers

Lemon

2 min read

·

31-01-2025

1

0

Share

This worksheet provides a comprehensive exploration of piecewise functions, including definitions, graphing techniques, and solving real-world applications. Whether you're a student looking to master this concept or an educator seeking engaging materials, this resource offers a structured approach to understanding and applying piecewise functions.

What are Piecewise Functions?

A piecewise function is a function defined by multiple sub-functions, each applying to a specific interval of the input. Instead of a single formula covering the entire domain, a piecewise function uses different formulas for different parts of its domain. These intervals are often defined using inequalities. The key is that each input value corresponds to exactly one output value, adhering to the fundamental definition of a function.

Example:

Consider the function:

f(x) = {
  x + 2,  if x < 0
  x^2,    if x ≥ 0
}

This function behaves differently depending on the value of x. If x is less than 0, the function uses the formula x + 2. If x is 0 or greater, it uses the formula x².

Graphing Piecewise Functions

Graphing piecewise functions requires careful attention to the intervals. Each sub-function is graphed only within its designated interval. It is crucial to consider the endpoints of each interval to determine whether the point is included (closed circle) or excluded (open circle).

Steps to Graphing:

1. Identify the intervals: Determine the intervals of x for each sub-function.
2. Graph each sub-function: Graph each sub-function only within its specified interval.
3. Consider endpoints: Use closed circles (•) for endpoints included in the interval and open circles (◦) for endpoints excluded.
4. Connect the pieces: The graph will be composed of separate segments, each corresponding to a sub-function.

Worksheet Problems (with Solutions)

Problem 1:

Evaluate the function f(x) = { 2x + 1, if x ≤ 2; x² - 3, if x > 2 } at the following points:

• f(0) = ?
• f(2) = ?
• f(3) = ?
• f(-1) = ?

Solution 1:

• f(0) = 2(0) + 1 = 1 (using the first sub-function because 0 ≤ 2)
• f(2) = 2(2) + 1 = 5 (using the first sub-function because 2 ≤ 2)
• f(3) = 3² - 3 = 6 (using the second sub-function because 3 > 2)
• f(-1) = 2(-1) + 1 = -1 (using the first sub-function because -1 ≤ 2)

Problem 2:

Graph the function:

g(x) = {
  -x + 1, if x < 1
  2,       if 1 ≤ x ≤ 3
  x - 2,   if x > 3
}

Solution 2:

This graph will consist of three line segments. The first segment will be a line with a slope of -1 and y-intercept of 1, but only for x values less than 1 (open circle at x=1). The second segment will be a horizontal line at y=2 from x=1 (closed circle) to x=3 (closed circle). The third segment will be a line with a slope of 1 and y-intercept of -2, starting at x=3 (open circle). Remember to use open and closed circles correctly at the interval boundaries. (A visual graph would be ideal here, but Markdown limits such functionality).

Problem 3:

Write a piecewise function to represent the following scenario: A taxi charges $3 for the first mile and $2 for each additional mile.

Solution 3:

Let C(m) represent the cost for m miles.

C(m) = {
  $3,       if 0 < m ≤ 1
  $3 + $2(m - 1), if m > 1
}

This function charges $3 for the first mile and then adds $2 for each additional mile beyond the first.

This worksheet provides a foundation for understanding and working with piecewise functions. Remember to practice more problems to solidify your understanding. For further assistance or more complex examples, consult your textbook or a qualified instructor.