Homework 2.3 Piecewise Functions Answers

Homework 2.3: Piecewise Functions – A Comprehensive Guide to Understanding and Solving Problems

This comprehensive guide provides detailed explanations and solutions for a typical Homework 2.3 assignment focusing on piecewise functions. Piecewise functions, a cornerstone of algebra and precalculus, describe functions defined by multiple sub-functions across different intervals of their domain. Understanding them is crucial for further studies in calculus and related fields. This guide will not only provide answers but also delve into the underlying concepts, ensuring a thorough grasp of the topic. We'll cover evaluating piecewise functions, graphing them, and solving related equations and inequalities.

I. Understanding Piecewise Functions

A piecewise function is a function defined by multiple sub-functions, each applying to a specific interval of the input (domain). It's like a puzzle where different pieces (sub-functions) fit together to create a complete picture. The general form looks like this:

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

Here, g(x), h(x), i(x), etc., are different functions, each defined over its specific interval ([a, b), [b, c), [c, d], etc.). Note the use of open and closed intervals; careful attention to these is crucial for accurate evaluation.

II. Evaluating Piecewise Functions

Evaluating a piecewise function means finding the output (f(x)) for a given input (x). The key is to first identify which sub-function applies to the given x-value based on the defined intervals.

Example:

Let's say we have the piecewise function:

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

Now, let's evaluate f(1) and f(3):

• f(1): Since 1 < 2, we use the first sub-function: f(1) = (1)² + 1 = 2

• f(3): Since 3 ≥ 2, we use the second sub-function: f(3) = 3(3) - 1 = 8

III. Graphing Piecewise Functions

Graphing piecewise functions requires plotting each sub-function within its designated interval. Pay close attention to the endpoints of each interval – open circles (o) indicate that the endpoint is not included, while closed circles (•) indicate inclusion.

Example:

Let's graph the function from the previous example:

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

1. Graph x² + 1 for x < 2: This is a parabola. Plot points for x-values less than 2, remembering to use an open circle at x = 2 because the inequality is strictly less than.

2. Graph 3x - 1 for x ≥ 2: This is a straight line. Plot points for x-values greater than or equal to 2, using a closed circle at x = 2.

The resulting graph will show a parabola ending at an open circle at (2, 5) and a line starting at a closed circle at (2, 5). The two parts connect seamlessly at x=2, making the function continuous at this point. Not all piecewise functions are continuous, however.

IV. Solving Equations and Inequalities with Piecewise Functions

Solving equations or inequalities involving piecewise functions requires careful consideration of the different intervals. You'll need to solve the equation or inequality separately for each sub-function, ensuring that your solutions fall within the appropriate intervals.

Example:

Let's solve the equation f(x) = 5 for the piecewise function:

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

1. Solve x² + 1 = 5 for x < 2: This gives x² = 4, so x = ±2. However, we must discard x = -2 because it doesn't satisfy the condition x < 2. Therefore, x = 2 is a potential solution, but remember we’ll need to check this in the next step given that it's on the boundary between intervals.

2. Solve 3x - 1 = 5 for x ≥ 2: This gives 3x = 6, so x = 2. This solution satisfies the condition x ≥ 2.

Since both sub-functions lead to x=2, we need to analyze continuity at x =2. The function is indeed continuous since lim (x->2-) f(x) = lim (x->2+) f(x) = f(2) = 5. Therefore, the only solution is x = 2.

V. Common Mistakes to Avoid

Several common pitfalls can arise when working with piecewise functions:

• Ignoring interval boundaries: Pay close attention to whether the endpoints of the intervals are included or excluded (open vs. closed circles).

• Incorrectly identifying the appropriate sub-function: Always check which interval the input x-value falls into before evaluating the function.

• Neglecting to check for continuity: For some problems, especially those involving limits and derivatives (in later studies), understanding continuity at the boundaries is critical.

• Misinterpreting graphs: Carefully examine open and closed circles on the graph to correctly interpret the function's behavior.

VI. Advanced Applications of Piecewise Functions

Piecewise functions have many real-world applications across various disciplines, including:

• Modeling real-world phenomena: Piecewise functions are ideal for representing situations where the relationship between variables changes depending on the input's value. For example, tax brackets, delivery costs (with different rates for varying weight ranges), and progressive pricing models.

• Computer graphics: They are used to define shapes and images, allowing for precise control over curves and lines.

• Signal processing: Piecewise functions are essential for representing and analyzing signals that vary in different parts of their domain.

VII. Homework 2.3 Example Problems and Solutions (Illustrative)

Let's address some typical problems encountered in Homework 2.3, offering step-by-step solutions:

Problem 1:

Evaluate the following piecewise function at the given points:

f(x) = {  |x| + 2, if x < 0
          x² - 1, if 0 ≤ x ≤ 3
          √(x-3) + 2, if x > 3
}

Evaluate: f(-2), f(0), f(2), f(4), f(3)

Solution:

• f(-2): Since -2 < 0, f(-2) = |-2| + 2 = 4
• f(0): Since 0 ≤ 0 ≤ 3, f(0) = (0)² - 1 = -1
• f(2): Since 0 ≤ 2 ≤ 3, f(2) = (2)² - 1 = 3
• f(4): Since 4 > 3, f(4) = √(4-3) + 2 = 3
• f(3): Since 0 ≤ 3 ≤ 3, f(3) = (3)² - 1 = 8

Problem 2:

Graph the following piecewise function:

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

Solution:

This requires graphing three separate parts:

1. -x + 1 for x < 1: A line with a y-intercept of 1 and a slope of -1. Use an open circle at (1, 0).

2. 2 for 1 ≤ x ≤ 3: A horizontal line at y = 2, from x = 1 (closed circle) to x = 3 (closed circle).

3. x - 2 for x > 3: A line with a y-intercept of -2 and a slope of 1. Use an open circle at (3, 1).

Problem 3:

Solve the inequality f(x) > 0 for the piecewise function:

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

Solution:

1. Solve x + 2 > 0 for x ≤ -1: This gives x > -2. Combining this with the condition x ≤ -1, we get -2 < x ≤ -1.

2. Solve -x² > 0 for x > -1: This inequality has no solution since -x² is always less than or equal to 0, unless x = 0. However, -x² is strictly less than zero for all x except 0. Therefore, no solutions.

Combining both parts, the solution to f(x) > 0 is -2 < x ≤ -1.

VIII. Conclusion

Mastering piecewise functions is a vital step in your mathematical journey. By understanding the underlying concepts, practicing evaluation, graphing, and equation-solving techniques, and avoiding common errors, you can confidently tackle any homework assignment involving piecewise functions and build a strong foundation for future mathematical studies. Remember that consistent practice and attention to detail are key to success in this area. This guide provides a solid framework; continue practicing with various problems to reinforce your understanding and build confidence.