evaluating piecewise functions answer key

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12-01-2025

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Piecewise functions, those intriguing mathematical constructs defined by multiple sub-functions across different intervals, can initially seem daunting. However, with a structured approach and a clear understanding of the underlying principles, evaluating these functions becomes straightforward. This guide will walk you through the process, providing examples and an answer key to solidify your understanding.

Understanding Piecewise Functions

A piecewise function is defined by different formulas or expressions for different parts of its domain. The domain is divided into intervals, and each interval has a corresponding sub-function. The key to evaluating a piecewise function lies in identifying which sub-function applies to the input value (x).

General Form:

A piecewise function typically looks like this:

f(x) = {
  g(x),  if a ≤ x < b
  h(x),  if b ≤ x < c
  i(x),  if x ≥ c
}

Where:

• f(x) is the piecewise function.
• g(x), h(x), and i(x) are the sub-functions.
• a, b, and c define the intervals of the domain.

Crucial Step: Always determine which interval your input value (x) falls into before applying the corresponding sub-function.

Step-by-Step Evaluation Process

Let's illustrate the evaluation process with an example:

Example:

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

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

Solution:

1. x = -2: Since -2 < 0, we use the first sub-function: f(-2) = 2(-2) + 1 = -3

2. x = 0: Since 0 ≤ 0 ≤ 2, we use the second sub-function: f(0) = (0)² = 0

3. x = 3: Since 3 > 2, we use the third sub-function: f(3) = 5 - 3 = 2

Practice Problems with Answer Key

Now, let's test your understanding with some practice problems. Try evaluating the following piecewise functions for the given x values.

Problem 1:

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

Evaluate: f(-1), f(1), f(3)

Problem 2:

g(x) = {
  |x|,     if x < 2
  x - 1,   if 2 ≤ x ≤ 5
  10,      if x > 5
}

Evaluate: g(0), g(2), g(5), g(8)

Problem 3:

h(x) = {
  1/x,      if x < -1
  x²,        if -1 ≤ x ≤ 1
  √(x-1),   if x > 1
}

Evaluate: h(-2), h(0), h(1), h(5)

Answer Key

Problem 1:

• f(-1) = 2
• f(1) = 4
• f(3) = 5

Problem 2:

• g(0) = 0
• g(2) = 1
• g(5) = 4
• g(8) = 10

Problem 3:

• h(-2) = -1/2
• h(0) = 0
• h(1) = 1
• h(5) = 2

This comprehensive guide, along with the practice problems and answer key, should provide a solid foundation for evaluating piecewise functions. Remember, the key is accurately identifying the correct sub-function based on the input value's position within the defined intervals. Practice makes perfect, so keep working through examples to build your proficiency!