Piecewise functions are mathematical functions that are defined by different rules or formulas over different intervals. They are often used to model real-world situations where different rules apply in different scenarios. Understanding piecewise functions is essential in algebra and calculus, as they allow us to accurately represent complex relationships and make predictions based on different conditions.
When working with piecewise functions, it is important to identify the different rules or formulas that apply to each interval and to understand how to piece them together to form a complete function. This worksheet on piecewise functions is designed to help students practice working with these types of functions and become more comfortable with their properties and applications.
1. Given the piecewise function f(x) =
-3x + 2 if x < 1,
x^2 if x >= 1
, evaluate f(0), f(1), and f(2).
2. Graph the piecewise function g(x) =
2x + 1 if x < -1,
x – 1 if -1 <= x <= 1,
x^2 if x > 1
on a coordinate plane and identify the intervals where each rule applies.
3. Solve the inequality h(x) =
2x – 3 if x < 0,
x^2 – 4 if x >= 0
> 0 and express the solution in interval notation.
4. Write a piecewise function k(x) that models the following scenario: A company charges $10 per hour for the first 5 hours of service and $15 per hour for each additional hour thereafter.
5. Determine the domain and range of the piecewise function m(x) =
3x – 1 if x < 2,
x^2 if x >= 2
and explain your reasoning.
Overall, this worksheet on piecewise functions offers a variety of problems that will challenge students to think critically and apply their knowledge of functions and inequalities. By practicing with these types of functions, students will develop a deeper understanding of how to work with complex mathematical relationships and improve their problem-solving skills.