piecewise functions worksheet with answers pdf

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31-01-2025

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Mastering Piecewise Functions: A Comprehensive Worksheet with Answers

This worksheet provides a thorough exploration of piecewise functions, covering key concepts and various difficulty levels. It's designed to help students solidify their understanding and build problem-solving skills. Each problem includes a detailed solution, allowing for self-assessment and improved learning. Whether you're a student preparing for an exam or a teacher seeking supplemental materials, this resource offers a valuable tool for mastering piecewise functions.

What are Piecewise Functions?

A piecewise function is a function defined by multiple subfunctions, each applicable over a specific interval of the domain. Essentially, it's a function "broken" into pieces. The key to understanding and working with piecewise functions lies in correctly identifying the appropriate subfunction based on the input value (x).

Section 1: Evaluating Piecewise Functions

Instructions: Evaluate the following piecewise functions for the given values of x.

Problem 1:

f(x) =
{ 2x + 1, if x < 0 { x² - 3, if x ≥ 0

a) f(-2) = ? b) f(0) = ? c) f(3) = ?

Answer 1:

a) Since -2 < 0, we use the first subfunction: f(-2) = 2(-2) + 1 = -3 b) Since 0 ≥ 0, we use the second subfunction: f(0) = 0² - 3 = -3 c) Since 3 ≥ 0, we use the second subfunction: f(3) = 3² - 3 = 6

Problem 2:

g(x) =
{ |x| + 2, if x ≤ 1 { √(x-1), if x > 1

a) g(-1) = ? b) g(1) = ? c) g(5) = ?

Answer 2:

a) Since -1 ≤ 1, we use the first subfunction: g(-1) = |-1| + 2 = 3 b) Since 1 ≤ 1, we use the first subfunction: g(1) = |1| + 2 = 3 c) Since 5 > 1, we use the second subfunction: g(5) = √(5-1) = 2

Section 2: Graphing Piecewise Functions

Instructions: Graph the following piecewise functions. Remember to consider the domain restrictions for each subfunction.

Problem 3:

h(x) =
{ x + 1, if x < 2 { -x + 4, if x ≥ 2

Answer 3: (This requires a graph. The graph will show a line with a slope of 1 and y-intercept of 1 for x < 2, and a line with a slope of -1 and y-intercept of 4 for x ≥ 2. The two lines meet at the point (2, 2).) A visual representation would be ideal here if this were a true PDF; consider using graphing software like Desmos to create the visual.

Problem 4:

k(x) =
{ 1, if -2 ≤ x < 0 { x, if 0 ≤ x ≤ 2 { 3, if x > 2

Answer 4: (This also requires a graph. The graph will show a horizontal line at y=1 from x=-2 to x=0 (open circle at x=0), a diagonal line from (0,0) to (2,2), and a horizontal line at y=3 from x=2 onwards.) Again, a visual is necessary for a complete answer and would be included in a PDF version.

Section 3: Writing Piecewise Functions

Instructions: Write the piecewise function represented by the following descriptions or graphs. (Graphs would be included in a PDF version).

Problem 5:

A function that is equal to x² when x is less than 1 and equal to x + 1 when x is greater than or equal to 1.

Answer 5:

f(x) =
{ x², if x < 1 { x + 1, if x ≥ 1

Problem 6: (This problem would include a graph in the PDF)

(Description of a graph showing a piecewise function)

Answer 6: (The answer would be a piecewise function based on the provided graph.)

This worksheet provides a starting point. More complex problems involving absolute value, step functions, and applications of piecewise functions could be added for a more advanced level. Remember to always carefully consider the domain restrictions when working with piecewise functions. A PDF version would include the necessary graphs and offer a more visually appealing learning experience.