piecewise functions worksheet and answers

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11-01-2025

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This worksheet provides a comprehensive exploration of piecewise functions, including definitions, graphing techniques, and practical applications. Solutions are provided to allow for self-assessment and a deeper understanding of the concepts. This resource is ideal for students in algebra, pre-calculus, and calculus courses.

What are Piecewise Functions?

A piecewise function is a function defined by multiple subfunctions, each applicable over a specified interval of the domain. Essentially, it's a function "broken" into pieces. Each piece is defined by a separate equation, and the entire function is defined by combining these pieces. The key is understanding which equation to use based on the input value (x).

Key Features of Piecewise Functions:

• Multiple subfunctions: A piecewise function consists of two or more functions.
• Defined intervals: Each subfunction is defined only over a specific interval or range of x-values.
• Transition points: The points where the intervals meet are called transition points and are crucial for graphing and evaluating the function.
• Domain: The domain of a piecewise function is the union of the intervals of its subfunctions.

Graphing Piecewise Functions

Graphing piecewise functions requires careful attention to the intervals and the behavior of each subfunction within its defined range. Here's a step-by-step approach:

1. Identify the subfunctions and their respective intervals. Pay close attention to the inequalities defining each interval (e.g., x < 2, x ≥ 2).
2. Graph each subfunction separately within its defined interval. Use the appropriate techniques for graphing each subfunction (linear, quadratic, etc.). Pay attention to whether the endpoint is included (closed circle) or excluded (open circle) based on the inequality.
3. Combine the graphs. Bring all the individual graphs together onto a single coordinate plane.

Example Problems and Solutions

Let's work through a few examples to solidify our understanding.

Problem 1:

Graph the piecewise function:

f(x) = {
  x + 2,  x < 1
  x^2,   x ≥ 1
}

Solution 1:

1. Subfunctions and intervals: We have two subfunctions: f(x) = x + 2 for x < 1 and f(x) = x^2 for x ≥ 1.
2. Graphing separately:
   • x + 2 is a line with a slope of 1 and a y-intercept of 2. We graph this line only for x-values less than 1 (open circle at x = 1).
   • x^2 is a parabola. We graph this for x-values greater than or equal to 1 (closed circle at x = 1).
3. Combining the graphs: The complete graph shows a line segment to the left of x = 1 and a parabola to the right of and including x = 1.

Problem 2:

Evaluate the piecewise function:

g(x) = {
  3x - 1, x ≤ 0
  2,       0 < x < 3
  x + 1,   x ≥ 3
}

at x = -2, x = 1, and x = 5.

Solution 2:

• g(-2): Since -2 ≤ 0, we use the first subfunction: g(-2) = 3(-2) - 1 = -7
• g(1): Since 0 < 1 < 3, we use the second subfunction: g(1) = 2
• g(5): Since 5 ≥ 3, we use the third subfunction: g(5) = 5 + 1 = 6

Practice Problems

1. Graph the piecewise function:

   h(x) = {
     |x|,     x < 2
     -x + 4, x ≥ 2
   }
   

2. Evaluate the piecewise function:

   k(x) = {
     x^2 - 1, x ≤ -1
     1/x,      -1 < x < 1
     √x,       x ≥ 1
   }
   

   at x = -3, x = 0, and x = 4.

Answers to Practice Problems (Solutions will be provided upon request)

This worksheet offers a structured approach to understanding and working with piecewise functions. Remember to practice various examples and explore different types of subfunctions to master this concept. By working through these problems and understanding the solutions, you will strengthen your understanding of piecewise functions and their applications.