1.1 Day 2 Evaluating Piecewise Functions Answer Key

1.1 Day 2: Evaluating Piecewise Functions – A Comprehensive Guide

Evaluating piecewise functions can initially seem daunting, but with a structured approach and a clear understanding of the underlying concepts, it becomes significantly easier. This comprehensive guide will delve into the intricacies of evaluating piecewise functions, providing a detailed explanation, numerous examples, and a step-by-step process to master this essential mathematical skill. We'll move beyond simple answers and explore the underlying logic, empowering you to confidently tackle any piecewise function problem.

Understanding Piecewise Functions

A piecewise function is a function defined by multiple sub-functions, each applying to a specific interval of the domain. Think of it as a collection of different functions stitched together to create a single, albeit complex, function. Each sub-function has its own defined domain, ensuring that there's no overlap between them. This is crucial for the function to be well-defined.

The general form of a piecewise function is:

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

Where:

• f(x) represents the overall piecewise function.
• g(x), h(x), and k(x) are the individual sub-functions.
• a, b, c, and d define the intervals (domains) for each sub-function. Note that the intervals are often non-overlapping, ensuring that each input x corresponds to only one output f(x).

Step-by-Step Guide to Evaluating Piecewise Functions

The key to evaluating piecewise functions lies in identifying the correct sub-function to use based on the input value (x). Follow these steps:

1. Identify the Input: Determine the value of x you need to evaluate the function for.

2. Determine the Appropriate Sub-function: Examine the intervals defined for each sub-function in the piecewise definition. Find the interval that contains your input value x. Pay close attention to the inequality signs (≤, <, >, ≥). A number might belong to one interval and not another, so precise analysis is crucial.

3. Substitute and Evaluate: Once you've identified the correct sub-function, substitute the input value (x) into that specific sub-function and perform the necessary calculations.

4. Obtain the Output: The result of the calculation is the output f(x) for the given input x.

Examples: Evaluating Piecewise Functions

Let's illustrate the process with several examples of increasing complexity.

Example 1: A Simple Piecewise Function

Consider the piecewise function:

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

a) Evaluate f(1):

1. Input: x = 1
2. Sub-function: Since 1 < 2, we use the sub-function 2x + 1.
3. Substitution and Evaluation: f(1) = 2(1) + 1 = 3
4. Output: f(1) = 3

b) Evaluate f(4):

1. Input: x = 4
2. Sub-function: Since 4 ≥ 2, we use the sub-function x² - 3.
3. Substitution and Evaluation: f(4) = (4)² - 3 = 16 - 3 = 13
4. Output: f(4) = 13

Example 2: A Piecewise Function with More Intervals

Let's consider a more complex piecewise function:

g(x) = {
  x + 5,  if x ≤ -1
  x²,     if -1 < x < 2
  3x - 1, if x ≥ 2
}

a) Evaluate g(-2):

1. Input: x = -2
2. Sub-function: Since -2 ≤ -1, we use the sub-function x + 5.
3. Substitution and Evaluation: g(-2) = (-2) + 5 = 3
4. Output: g(-2) = 3

b) Evaluate g(0):

1. Input: x = 0
2. Sub-function: Since -1 < 0 < 2, we use the sub-function x².
3. Substitution and Evaluation: g(0) = (0)² = 0
4. Output: g(0) = 0

c) Evaluate g(3):

1. Input: x = 3
2. Sub-function: Since 3 ≥ 2, we use the sub-function 3x - 1.
3. Substitution and Evaluation: g(3) = 3(3) - 1 = 8
4. Output: g(3) = 8

Example 3: Handling Absolute Value in Piecewise Functions

Piecewise functions often involve absolute value functions, adding another layer of complexity. Let's consider:

h(x) = {
  |x| + 2, if x < 0
  x - 1,   if x ≥ 0
}

a) Evaluate h(-3):

1. Input: x = -3
2. Sub-function: Since -3 < 0, we use the sub-function |x| + 2.
3. Substitution and Evaluation: h(-3) = |-3| + 2 = 3 + 2 = 5
4. Output: h(-3) = 5

b) Evaluate h(0):

1. Input: x = 0
2. Sub-function: Since 0 ≥ 0, we use the sub-function x - 1.
3. Substitution and Evaluation: h(0) = 0 - 1 = -1
4. Output: h(0) = -1

Example 4: Piecewise Function with a Constant Sub-Function

Piecewise functions can also include constant sub-functions. Consider this example:

k(x) = {
  5,       if x ≤ 1
  2x + 1,  if x > 1
}

a) Evaluate k(1):

1. Input: x = 1
2. Sub-function: Since 1 ≤ 1, we use the sub-function 5 (a constant).
3. Substitution and Evaluation: k(1) = 5
4. Output: k(1) = 5

b) Evaluate k(3):

1. Input: x = 3
2. Sub-function: Since 3 > 1, we use the sub-function 2x + 1.
3. Substitution and Evaluation: k(3) = 2(3) + 1 = 7
4. Output: k(3) = 7

Common Mistakes to Avoid

• Incorrect Interval Selection: Carefully examine the inequalities defining each interval. A minor error in understanding "<" versus "≤" can lead to selecting the wrong sub-function and an incorrect result.

• Misinterpretation of Absolute Value: Remember that |x| = x if x ≥ 0 and |x| = -x if x < 0. Failure to account for this can lead to incorrect calculations.

• Order of Operations: Follow the standard order of operations (PEMDAS/BODMAS) when evaluating the sub-functions, ensuring accurate calculations.

Advanced Applications of Piecewise Functions

Piecewise functions are not merely abstract mathematical constructs; they find extensive applications in various fields, including:

• Modeling Real-world Phenomena: Piecewise functions effectively model situations where the relationship between variables changes abruptly at specific points. For example, the cost of a phone plan might have different rates depending on the number of minutes used.

• Computer Graphics: They are instrumental in creating various shapes and patterns in computer graphics, allowing for precise control over the curves and lines in an image.

• Economics and Finance: Piecewise functions can model tax brackets, where the tax rate changes depending on income level.

• Physics and Engineering: They can describe scenarios with discontinuous changes, such as the velocity of an object undergoing sudden acceleration or deceleration.

Conclusion

Mastering the evaluation of piecewise functions is a fundamental skill in mathematics. With a systematic approach, careful attention to detail, and practice with diverse examples, you can confidently navigate the complexities of these functions and apply them to real-world problem-solving. Remember to always double-check your interval selections and calculations to avoid common pitfalls. The process is straightforward once you understand the core principles. Through consistent practice, evaluating piecewise functions will become second nature, empowering you to tackle even the most challenging mathematical problems.