1.3 Piecewise Functions Homework Answers

Decoding Piecewise Functions: A Comprehensive Guide to Homework Problems (1.3)

Piecewise functions, often a source of confusion for students first encountering them, are actually quite elegant once you understand their underlying logic. This comprehensive guide will walk you through the intricacies of piecewise functions, providing clear explanations, worked examples, and strategies to tackle those tricky homework problems in section 1.3 (or equivalent). We'll cover evaluating functions, graphing them, and understanding their real-world applications. By the end, you'll be confident in your ability to conquer any piecewise function challenge.

Understanding Piecewise Functions: The Big Picture

A piecewise function is, simply put, a function defined by multiple sub-functions, each applicable over a specific interval of the domain. Imagine it like a road with different speed limits depending on the section you're driving through. Each "section" is defined by a sub-function and its associated interval. The key is understanding which sub-function to use based on the input value (x). The general form is represented as:

f(x) = {  g(x)  if  a ≤ x < b
          h(x)  if  b ≤ x < c
          i(x)  if  c ≤ x ≤ d
          ...and so on }

Here, g(x), h(x), i(x), etc., represent different functions, and a, b, c, d, etc., define the intervals over which each sub-function applies. Note the careful use of inequalities – this ensures that each x-value falls into only one defined interval, avoiding ambiguity.

Evaluating Piecewise Functions: Finding the Right Sub-Function

The most basic task is evaluating a piecewise function for a given x-value. The crucial step is identifying the correct sub-function based on the interval containing that x-value.

Example 1:

Let's say we have the following piecewise function:

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

Let's evaluate f(-1), f(1), and f(3):

• f(-1): Since -1 < 0, we use the first sub-function: f(-1) = 2(-1) + 1 = -1.
• f(1): Since 0 ≤ 1 ≤ 2, we use the second sub-function: f(1) = 1² = 1.
• f(3): Since 3 > 2, we use the third sub-function: f(3) = 5.

See? It's just about choosing the right tool for the job. The interval determines which sub-function we use.

Graphing Piecewise Functions: Visualizing the Separate Pieces

Graphing piecewise functions requires graphing each sub-function within its specified interval. This often results in a graph with distinct "pieces" or sections. Pay close attention to the endpoints of each interval. Will the point be included (closed circle) or excluded (open circle)? This is determined by the inequality used to define the interval.

Example 2:

Let's graph the function from Example 1:

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

1. Graph 2x + 1 for x < 0: This is a line with a slope of 2 and a y-intercept of 1. We graph this line for all x-values less than 0. We use an open circle at (0,1) because x=0 is not included in this interval.

2. Graph x² for 0 ≤ x ≤ 2: This is a parabola. We graph this section from x = 0 to x = 2, inclusive. We use closed circles at (0,0) and (2,4) because these points are included in the interval.

3. Graph 5 for x > 2: This is a horizontal line at y = 5. We graph this line for all x-values greater than 2. We use an open circle at (2,5) since x=2 is not included in this interval.

Combining these three sections gives us the complete graph of the piecewise function, clearly showing its distinct parts.

Solving Equations Involving Piecewise Functions: A Multi-Step Process

Solving equations involving piecewise functions requires careful consideration of each sub-function and its corresponding interval. You must determine which sub-function is relevant based on the potential solution.

Example 3:

Solve the equation f(x) = 4, where f(x) is the function from Example 1:

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

We need to solve 4 = 2x + 1, 4 = x², and 4 = 5.

• 4 = 2x + 1: Solving gives x = 3/2 = 1.5. Since 0 ≤ 1.5 ≤ 2, this is a valid solution.

• 4 = x²: Solving gives x = ±2. Since 0 ≤ 2 ≤ 2, x = 2 is a valid solution. x = -2 is not a valid solution because it doesn't fall within the interval 0 ≤ x ≤ 2.

• 4 = 5: This equation has no solution.

Therefore, the solutions to f(x) = 4 are x = 1.5 and x = 2.

Advanced Piecewise Functions: Absolute Value and Other Challenges

Absolute value functions are frequently incorporated into piecewise functions. Remember the definition of absolute value: |x| = x if x ≥ 0, and |x| = -x if x < 0. This allows you to rewrite absolute value expressions as piecewise functions.

Example 4:

Rewrite the absolute value function g(x) = |x - 2| as a piecewise function.

Since |x - 2| is 0 when x = 2, we can define the piecewise function as:

g(x) = { x - 2  if  x ≥ 2
         -(x - 2) if  x < 2 }

This simplifies to:

g(x) = { x - 2  if  x ≥ 2
         2 - x  if  x < 2 }

Real-World Applications: Seeing Piecewise Functions in Action

Piecewise functions aren't just abstract mathematical concepts; they have numerous practical applications:

• Tax brackets: Income tax systems often use piecewise functions to calculate tax owed based on income levels.
• Shipping costs: Shipping costs frequently depend on the weight of the package, with different rates applying to different weight ranges.
• Cell phone plans: The cost of a cell phone plan may vary depending on the amount of data used.
• Electricity pricing: Electricity companies often charge different rates for different usage levels.

Understanding piecewise functions is crucial to modeling these real-world scenarios accurately.

Frequently Asked Questions (FAQs)

• Q: What happens if an x-value falls on the boundary between two intervals?

  • A: The convention is to usually use the function defined by the closed interval. However, it's crucial to check the definition of the piecewise function for clarification. Sometimes, the function might be undefined at boundary points.

• Q: Can a piecewise function be continuous?

  • A: Yes. A piecewise function is continuous if the sub-functions are continuous within their intervals, and the function values at the boundaries match (no jumps).

• Q: Can a piecewise function be differentiable?

  • A: Yes, but it requires that the sub-functions are differentiable within their intervals, and the derivatives match at the boundaries (no sharp corners).

• Q: How do I determine if a piecewise function is even or odd?

  • A: There isn't a simple trick. You need to examine the symmetry of each sub-function within its interval and verify if the entire piecewise function meets the even or odd function criteria. In most cases, a piecewise function will neither be even nor odd.

Conclusion: Mastering the Art of Piecewise Functions

Piecewise functions, although initially seeming daunting, become manageable with practice and a systematic approach. By understanding the core principles – identifying the correct sub-function, graphing the individual pieces, and solving equations within the appropriate intervals – you'll be well-equipped to tackle any piecewise function problem. Remember to carefully examine the intervals and endpoints, and don't hesitate to break down complex problems into smaller, more manageable steps. With consistent effort, you'll master this essential mathematical concept and appreciate its power in modeling real-world situations. Remember to always double-check your work and use online graphing tools to visualize your functions and solutions, confirming your understanding. Good luck with your homework!