Math 2 Piecewise Functions Worksheet: Practice And Solutions
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Math worksheets are an essential part of a student's learning experience, especially when it comes to understanding complex topics such as piecewise functions. Piecewise functions are a unique way of defining a function using different expressions over different intervals. They can seem challenging at first, but with practice, they become easier to understand and apply. In this article, we will explore piecewise functions, their significance, and provide a comprehensive worksheet with practice problems and solutions.
Understanding Piecewise Functions
Piecewise functions are defined by different formulas depending on the input value. These functions can be expressed in a variety of ways, including absolute values, intervals, and step functions. They are particularly useful in real-world applications where different scenarios need to be described.
Definition and Notation
A piecewise function is usually written in the following format:
[ f(x) = \begin{cases} f_1(x) & \text{if } x < a \ f_2(x) & \text{if } a \leq x < b \ f_3(x) & \text{if } x \geq b \end{cases} ]
Where:
- ( f_1(x), f_2(x), f_3(x) ) are the different expressions used for each interval.
- ( a ) and ( b ) are the points that define the intervals.
Why are Piecewise Functions Important?
Piecewise functions model many real-life situations, such as tax brackets, shipping costs, and physics problems like the motion of objects with varying acceleration. Understanding how to work with these functions helps students grasp broader mathematical concepts and their applications in different fields.
Piecewise Functions Worksheet
To reinforce understanding, here’s a worksheet containing practice problems on piecewise functions. The worksheet is designed to test students’ abilities to evaluate, graph, and create piecewise functions.
Practice Problems
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Evaluate the function: [ f(x) = \begin{cases} 2x + 3 & \text{if } x < 1 \ x^2 & \text{if } 1 \leq x < 3 \ 5 & \text{if } x \geq 3 \end{cases} ] Find ( f(0), f(2), f(3) ).
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Graph the function: Use the following piecewise function to sketch its graph. [ g(x) = \begin{cases} -x + 4 & \text{if } x < 0 \ x^2 & \text{if } 0 \leq x \leq 2 \ 2x - 4 & \text{if } x > 2 \end{cases} ]
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Create a piecewise function: Define a piecewise function ( h(x) ) that outputs the following:
- 10 if ( x < -1 )
- ( x + 2 ) if ( -1 \leq x < 1 )
- 0 if ( x \geq 1 )
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Word problem: A taxi company charges $5 for the first mile and $2 for each additional mile. Write a piecewise function ( T(d) ) for the total cost of the taxi ride depending on the distance ( d ) traveled.
Solutions to Practice Problems
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Evaluating ( f(x) ):
- For ( f(0) ): [ f(0) = 2(0) + 3 = 3 ]
- For ( f(2) ): [ f(2) = 2^2 = 4 ]
- For ( f(3) ): [ f(3) = 5 ]
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Graphing ( g(x) ):
- For ( x < 0 ), ( g(x) = -x + 4 ) (a decreasing line).
- For ( 0 \leq x \leq 2 ), ( g(x) = x^2 ) (a parabola).
- For ( x > 2 ), ( g(x) = 2x - 4 ) (an increasing line).
The resulting graph should consist of three distinct segments connecting at ( x = 0 ) and ( x = 2 ).
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Piecewise function ( h(x) ): [ h(x) = \begin{cases} 10 & \text{if } x < -1 \ x + 2 & \text{if } -1 \leq x < 1 \ 0 & \text{if } x \geq 1 \end{cases} ]
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Cost function ( T(d) ): [ T(d) = \begin{cases} 5 & \text{if } d < 1 \ 5 + 2(d - 1) & \text{if } d \geq 1 \end{cases} ] This function reflects the initial charge plus the incremental charge for distances beyond the first mile.
Conclusion
Piecewise functions can be a powerful tool in both mathematics and real-world applications. By practicing with a variety of problems and understanding their structure, students can gain confidence and proficiency. Keep these worksheets handy and revisit them often to strengthen your skills with piecewise functions! 🎉🧮