piecewise functions worksheet answer key

3 min read 11-01-2025
piecewise functions worksheet answer key

This answer key provides solutions and detailed explanations for a typical piecewise function worksheet. Remember that specific questions and their solutions will vary depending on the worksheet itself. This guide will equip you to solve a wide range of piecewise function problems.

What are Piecewise Functions?

Piecewise functions are defined by different formulas for different parts of their domain. Think of them as a collection of individual functions stitched together. Each "piece" is defined over a specific interval. Understanding the domain restrictions is crucial for correctly evaluating these functions.

Example Problems & Solutions

Let's work through some common piecewise function problems. We'll illustrate how to evaluate the function at specific points and how to graph it.

Problem 1: Evaluating a Piecewise Function

Consider the piecewise function:

f(x) = 
  2x + 1, if x < 0
  x² - 3, if 0 ≤ x ≤ 2
  5, if x > 2

Find:

a) f(-2) b) f(0) c) f(1) d) f(3)

Solution 1:

a) Since -2 < 0, we use the first piece: f(-2) = 2(-2) + 1 = -3

b) Since 0 ≥ 0, we use the second piece: f(0) = (0)² - 3 = -3

c) Since 0 ≤ 1 ≤ 2, we use the second piece: f(1) = (1)² - 3 = -2

d) Since 3 > 2, we use the third piece: f(3) = 5

Problem 2: Graphing a Piecewise Function

Graph the piecewise function from Problem 1.

Solution 2:

To graph this function, consider each piece separately:

  • 2x + 1, if x < 0: This is a line with a slope of 2 and a y-intercept of 1. Graph this line for all x-values less than 0. Use an open circle at x = 0 to indicate that the point (0,1) is not included.

  • x² - 3, if 0 ≤ x ≤ 2: This is a parabola. Plot points for x = 0, 1, and 2 to get the points (0, -3), (1, -2), and (2, 1). Connect these points with a smooth curve.

  • 5, if x > 2: This is a horizontal line at y = 5. Graph this line for all x-values greater than 2. Use an open circle at x = 2 to indicate that the point (2,5) is not included.

Combine all three pieces on a single graph. The graph will show a line, a portion of a parabola, and a horizontal line.

Problem 3: Finding the Domain and Range

Determine the domain and range of the piecewise function from Problem 1.

Solution 3:

  • Domain: The domain is all real numbers, as each piece is defined for a specific interval covering all x-values from negative infinity to positive infinity. (−∞, ∞)

  • Range: The range is (-∞, 1) ∪ [-3, 5]. The range excludes values between 1 and -3 (exclusive of 1) and includes all values greater than or equal to -3 up to and including 5.

Problem 4: Writing a Piecewise Function

Write a piecewise function for the graph shown below (Assume a graph is provided with distinct segments).

Solution 4:

This problem requires you to carefully examine the provided graph. Determine the equation for each segment and the intervals where those equations are valid. For example, if one segment is a straight line, find its slope and y-intercept to write its equation in slope-intercept form (y = mx + b). If a segment is a parabola, determine its vertex and a point on the curve to find its equation in vertex form or standard form.

Tips for Solving Piecewise Function Problems

  • Identify the intervals: Carefully determine the intervals where each piece of the function is defined.
  • Substitute correctly: When evaluating, use the correct piece of the function based on the input value.
  • Pay attention to endpoints: Remember that open or closed circles at endpoints indicate whether the endpoint is included in the interval.
  • Graph each piece separately: When graphing, graph each piece of the function on its respective interval.

This answer key and explanation should provide a strong foundation for understanding and solving piecewise function problems. Remember to practice with various examples to solidify your understanding. Remember to consult your specific worksheet for the exact problems and their solutions.

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