evaluating piecewise functions worksheet with answers pdf

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03-02-2025

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Evaluating Piecewise Functions: A Comprehensive Worksheet with Answers

This worksheet provides a thorough exploration of evaluating piecewise functions, a crucial concept in algebra and precalculus. We'll cover various function types and complexities, building your understanding step-by-step. Each problem includes a detailed solution, allowing for self-assessment and a strong grasp of the material.

What are Piecewise Functions?

A piecewise function is a function defined by multiple subfunctions, each applicable over a specific interval of the domain. These intervals are mutually exclusive, meaning no two intervals overlap. Think of it like a function with multiple "pieces" – each piece is a different function, active only within its assigned domain segment. The function's behavior changes depending on the input value's location within the defined intervals.

The general form of a piecewise function is:

f(x) = {  g(x), if x ∈ A
          h(x), if x ∈ B
          i(x), if x ∈ C
          ...
}

Where g(x), h(x), i(x)... are different functions, and A, B, C... are disjoint intervals forming the function's domain.

Evaluating Piecewise Functions: A Step-by-Step Approach

Evaluating a piecewise function requires two steps:

1. Identify the Correct Subfunction: Determine which interval the input value (x) belongs to. This dictates which subfunction you'll use to calculate the output (f(x)).

2. Substitute and Evaluate: Substitute the input value into the chosen subfunction and perform the necessary calculations to find the output.

Example Problem 1:

Let's consider the following piecewise function:

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

a) Evaluate f(-2):

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

b) Evaluate f(3):

Since 3 ≥ 0, we use the second subfunction: f(3) = 3² = 9

Example Problem 2 (More Complex):

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

a) Evaluate g(0):

0 ≤ 2, so we use the first subfunction: g(0) = |0 - 1| = 1

b) Evaluate g(6):

6 > 2, so we use the second subfunction: g(6) = √(6 - 2) = √4 = 2

Worksheet Problems (with Answers at the end):

1. Evaluate h(x) = { x + 5, if x < 3; -2x + 1, if x ≥ 3 } for h(-1), h(3), and h(5).

2. Evaluate k(x) = { 1/(x-2), if x ≠ 2; 0, if x = 2 } for k(0), k(2), and k(4).

3. Evaluate p(x) = { x³, if x ≤ -1; 2x + 3, if -1 < x < 2; -x + 7, if x ≥ 2 } for p(-2), p(0), and p(3).

4. Evaluate m(x) = { √(9-x²), if -3 ≤ x ≤ 3; 0, otherwise } for m(-3), m(0), and m(4).

Answers:

1. h(-1) = 4, h(3) = -5, h(5) = -9

2. k(0) = -1/2, k(2) = 0, k(4) = 1/2

3. p(-2) = -8, p(0) = 3, p(3) = 4

4. m(-3) = 0, m(0) = 3, m(4) = 0

This worksheet provides a solid foundation for understanding and evaluating piecewise functions. Remember to always check which interval your input belongs to before performing the calculation. Further practice with more complex examples will solidify your understanding.