evaluating piecewise functions answer key

CimarronTexas Editorial Teams

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11-01-2025

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Piecewise functions, those mathematical chameleons that change their behavior depending on the input, can seem daunting at first. But with a systematic approach and a solid understanding of their structure, evaluating them becomes straightforward. This guide will walk you through the process, providing clear explanations and example problems with a complete answer key. We'll cover various scenarios, including those involving absolute values and inequalities, ensuring you're well-equipped to tackle any piecewise function problem.

Understanding Piecewise Functions

A piecewise function is defined by multiple sub-functions, each applicable over a specific interval of the domain. The key is identifying which sub-function to use based on the input value (often denoted by 'x'). The general form looks like this:

f(x) = {  f₁(x), if x ∈ I₁
          f₂(x), if x ∈ I₂
          f₃(x), if x ∈ I₃
          ...
        }

Where:

• f₁(x), f₂(x), f₃(x)... are different functions.
• I₁, I₂, I₃... are intervals defining the domains for each sub-function. These intervals can be open, closed, or half-open.

Step-by-Step Evaluation Process

To evaluate a piecewise function at a specific point, follow these steps:

1. Identify the Interval: Determine which interval the input value falls into.
2. Select the Correct Sub-function: Choose the sub-function associated with that interval.
3. Substitute and Evaluate: Substitute the input value into the chosen sub-function and calculate the result.

Example Problems with Answer Key

Let's work through some examples to solidify your understanding.

Example 1:

Evaluate the following piecewise function at x = 2, x = 0, and x = -3:

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

Solution:

• f(2): Since 2 > 1, we use the first sub-function: f(2) = 2² + 1 = 5
• f(0): Since -1 ≤ 0 ≤ 1, we use the second sub-function: f(0) = 2(0) = 0
• f(-3): Since -3 < -1, we use the third sub-function: f(-3) = -3 - 3 = -6

Example 2 (Involving Absolute Value):

Evaluate the following piecewise function at x = 3, x = 0, and x = -2:

g(x) = {  |x| + 2, if x ≥ 0
          -x,      if x < 0
        }

Solution:

• g(3): Since 3 ≥ 0, we use the first sub-function: g(3) = |3| + 2 = 5
• g(0): Since 0 ≥ 0, we use the first sub-function: g(0) = |0| + 2 = 2
• g(-2): Since -2 < 0, we use the second sub-function: g(-2) = -(-2) = 2

Example 3 (More Complex Intervals):

Evaluate h(x) at x = 1, x = 4, and x = 7:

h(x) = { x/2, if 0 < x ≤ 2
         x -1, if 2 < x ≤ 5
         √(x-5), if x > 5
       }

Solution:

• h(1): Since 0 < 1 ≤ 2, h(1) = 1/2 = 0.5
• h(4): Since 2 < 4 ≤ 5, h(4) = 4 - 1 = 3
• h(7): Since 7 > 5, h(7) = √(7-5) = √2

Answer Key Summary:

This guide provides a robust foundation for evaluating piecewise functions. Remember to always carefully examine the intervals defined for each sub-function before substituting your input value. Practice with different examples, and you'll master this essential mathematical concept in no time.