1.4 Piecewise Functions Homework Answers

Mastering Piecewise Functions: A Comprehensive Guide with Homework Solutions

Piecewise functions, those mathematical chameleons that change their behavior depending on the input, can initially seem daunting. However, with a structured approach and a clear understanding of their components, mastering piecewise functions becomes achievable and even enjoyable. This comprehensive guide will delve into the intricacies of piecewise functions, providing detailed explanations, worked examples, and solutions to common homework problems. We'll cover everything from understanding the notation to tackling complex scenarios, ensuring you develop a firm grasp of this important mathematical concept.

Understanding Piecewise Function Notation

A piecewise function is defined by multiple sub-functions, each applicable over a specific interval of the domain. The notation clearly outlines these sub-functions and their respective domains. Let's break down the typical structure:

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

In this example:

• f(x) represents the overall piecewise function.
• g(x), h(x), and k(x) are the individual sub-functions.
• a, b, and c define the intervals or domains over which each sub-function applies. Notice the use of inequalities to specify the intervals; this is crucial for accurate evaluation. Pay close attention to whether the endpoints are included (≤) or excluded (<).

Key takeaway: Understanding the notation is paramount. Each sub-function is only valid within its specified domain.

Evaluating Piecewise Functions: A Step-by-Step Approach

Evaluating a piecewise function at a given point involves two key steps:

1. Identify the Correct Sub-function: Determine which interval the input value (x) falls into.

2. Substitute and Evaluate: Substitute the input value into the corresponding sub-function and calculate the result.

Let's illustrate with an example:

Consider the piecewise function:

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

Example 1: Find f(1)

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

Example 2: Find f(2)

1. Since 2 ≥ 2, we use the second sub-function: x² - 1.
2. Substitute x = 2: 2² - 1 = 3. Therefore, f(2) = 3.

Example 3: Find f(3)

1. Since 3 ≥ 2, we use the second sub-function: x² - 1.
2. Substitute x = 3: 3² - 1 = 8. Therefore, f(3) = 8.

Example 4: Find f(-1)

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

Graphing Piecewise Functions: Visualizing the Behavior

Graphing piecewise functions helps visualize their behavior across different intervals. The process involves graphing each sub-function within its specified domain. Pay close attention to the endpoints:

• An open circle (◦) indicates an endpoint is excluded.
• A closed circle (•) indicates an endpoint is included.

Let's graph the example function from above:

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

1. Graph 2x + 1 for x < 2: This is a line with a slope of 2 and a y-intercept of 1. Start at (0,1) and draw a line with a slope of 2, but stop at x = 2. Place an open circle at (2,5) since x=2 is not included in this interval.

2. Graph x² - 1 for x ≥ 2: This is a parabola. Start at (2,3) with a closed circle (because x=2 is included) and continue the parabola to the right.

The resulting graph will show a line segment ending at an open circle at (2,5) and a parabola starting at a closed circle at (2,3).

Solving Piecewise Function Equations

Solving equations involving piecewise functions requires careful consideration of the domains. You must determine which sub-function is relevant based on the potential solutions. Let's consider an example:

Solve for x: f(x) = 3, where f(x) is defined as:

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

1. Consider the first sub-function: 2x + 1 = 3. This gives x = 1. Since 1 < 2, this solution is valid.

2. Consider the second sub-function: x² - 1 = 3. This gives x² = 4, so x = ±2. Since x ≥ 2, x = 2 is a valid solution. x = -2 is not valid because it doesn't fall within the domain of the second sub-function.

Therefore, the solutions are x = 1 and x = 2.

Advanced Piecewise Function Problems: Absolute Value Functions

Absolute value functions are frequently expressed as piecewise functions. Recall that |x| = x if x ≥ 0, and |x| = -x if x < 0. This allows us to rewrite absolute value functions as piecewise functions, simplifying analysis and solving equations.

Example: Rewrite |x - 2| as a piecewise function.

• If x - 2 ≥ 0 (meaning x ≥ 2), then |x - 2| = x - 2.
• If x - 2 < 0 (meaning x < 2), then |x - 2| = -(x - 2) = 2 - x.

Therefore, the piecewise function is:

f(x) = { 2 - x, if x < 2
          x - 2, if x ≥ 2 }

Homework Problem Solutions (1.4 Piecewise Functions)

To provide specific solutions to your homework, please provide the actual problems from section 1.4 of your textbook. I need the specific piecewise functions and questions to give you accurate and detailed answers.

However, I can offer a few example problems and solutions to illustrate the concepts further:

Problem 1:

Given the piecewise function:

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

Find: a) f(-2), b) f(1), c) f(3)

Solutions:

a) Since -2 ≤ 1, use the first sub-function: f(-2) = -2 + 3 = 1 b) Since 1 ≤ 1, use the first sub-function: f(1) = 1 + 3 = 4 c) Since 3 > 1, use the second sub-function: f(3) = 2(3) - 1 = 5

Problem 2:

Solve for x: f(x) = 5, where

f(x) = { 3x - 2, if x ≤ 3
          x + 4, if x > 3 }

Solutions:

• First sub-function: 3x - 2 = 5 => 3x = 7 => x = 7/3. Since 7/3 ≤ 3 is false, this solution is invalid.

• Second sub-function: x + 4 = 5 => x = 1. Since 1 > 3 is false, this solution is also invalid.

Therefore, there are no solutions to f(x) = 5.

Problem 3:

Graph the following piecewise function:

f(x) = { -x, if x < 0
          x², if x ≥ 0 }

Solutions:

This involves graphing two parts:

• For x < 0, graph y = -x (a line with a slope of -1 passing through the origin. Use an open circle at (0,0)).

• For x ≥ 0, graph y = x² (a parabola starting at (0,0) with a closed circle).

Frequently Asked Questions (FAQ)

• Q: What happens if the intervals overlap in a piecewise function? A: Overlapping intervals are generally a mistake in the definition of the piecewise function. Each x-value should belong to only one sub-function's domain.

• Q: Can piecewise functions be continuous? A: Yes, a piecewise function can be continuous if the values of the sub-functions match at the boundaries of their intervals.

• Q: How do I find the domain of a piecewise function? A: The domain is the union of all the intervals specified for each sub-function.

• Q: Can piecewise functions be differentiable? A: A piecewise function is differentiable if it is continuous and the derivatives of the sub-functions match at the boundaries.

Conclusion

Piecewise functions are a powerful tool in mathematics, enabling us to model systems with different behaviors across various intervals. By understanding the notation, evaluating functions, graphing them, and solving equations involving piecewise functions, you gain a crucial skill set applicable across various mathematical fields. Remember to always carefully consider the domain of each sub-function when working with these versatile functions. Don't hesitate to practice with diverse examples to build your confidence and mastery. Submit your specific homework problems, and I'll happily provide tailored solutions to help you succeed.