Piecewise functions, a cornerstone of Algebra 2, can initially seem daunting. However, with a structured approach and plenty of practice, mastering them becomes achievable. This guide provides a comprehensive worksheet, complete with examples and explanations, to solidify your understanding of piecewise functions and their applications.
What are Piecewise Functions?
A piecewise function is defined by multiple sub-functions, each applying to a specific interval of the domain. Think of it as a function made up of different pieces, each with its own rule. The key is understanding which rule to apply based on the input value (x).
General Form:
A piecewise function typically looks like this:
f(x) = {
g(x), if a ≤ x < b
h(x), if b ≤ x < c
i(x), if x ≥ c
}
Where:
f(x)represents the overall piecewise function.g(x),h(x), andi(x)are individual sub-functions.a,b, andcdefine the intervals or domains for each sub-function. These intervals are crucial for determining which sub-function to use.
Worksheet: Piecewise Functions
Instructions: For each problem, evaluate the piecewise function at the given x-value. Show your work and explain your reasoning.
Problem 1:
Given the piecewise function:
f(x) = {
2x + 1, if x < 2
x² - 3, if x ≥ 2
}
Evaluate:
a) f(1) b) f(2) c) f(4) d) f(0)
Problem 2:
Given the piecewise function:
g(x) = {
|x|, if x ≤ -1
x + 2, if -1 < x < 1
3, if x ≥ 1
}
Evaluate:
a) g(-2) b) g(0) c) g(1) d) g(-1)
Problem 3:
Graph the following piecewise function:
h(x) = {
-x + 1, if x ≤ 0
x + 1, if x > 0
}
Problem 4:
Write the piecewise function represented by the following graph: (Insert a graph here showing a piecewise function with distinct linear segments.)
Problem 5 (Challenge):
A cell phone company charges $20 per month for 500 minutes of talk time and $0.25 for each additional minute. Write a piecewise function that models the monthly cost, C(m), as a function of the number of minutes, m, used.
Solutions and Explanations
(Solutions will be provided here, offering step-by-step guidance and explanations for each problem. For Problem 3, a correctly graphed function will be shown. Problem 4 will have the piecewise function written out. Problem 5's solution will detail the creation of the piecewise function.)
This detailed approach will ensure a comprehensive understanding of piecewise functions. Remember, practice is key! Work through these problems carefully, and don't hesitate to consult your textbook or teacher for further assistance. Understanding piecewise functions is crucial for tackling more advanced concepts in Algebra 2 and beyond.
