Find The Range Of The Following Piecewise Function.

Finding the Range of Piecewise Functions: A Comprehensive Guide

Finding the range of a piecewise function can seem daunting at first, but with a systematic approach and a solid understanding of function behavior, it becomes a manageable task. This comprehensive guide will walk you through the process, providing examples and explanations to help you master this important concept in mathematics. We'll explore various types of piecewise functions and the strategies you can use to effectively determine their range. This guide will also touch upon the use of graphical methods and analytical techniques to solidify your understanding.

What is a Piecewise Function and its Range?

A piecewise function is a function defined by multiple sub-functions, each applicable to a specific interval of the domain. Essentially, it's a function "pieced together" from different parts. The range of a function, in general, is the set of all possible output values (y-values) the function can produce. For a piecewise function, the range is determined by considering the output values of each sub-function within its defined interval and then combining these ranges to find the overall range of the entire function.

Methods for Finding the Range of a Piecewise Function

There are two primary methods for finding the range of a piecewise function:

1. Graphical Method: This involves sketching the graph of the piecewise function and visually identifying the set of y-values covered by the graph.

2. Analytical Method: This involves analyzing the individual sub-functions and their respective domains to determine the range of each part, then combining these ranges to find the overall range.

Step-by-Step Guide: Analytical Approach

Let's break down the analytical approach into a step-by-step process:

1. Identify the Sub-functions and Their Domains: Carefully examine the piecewise function definition. Identify each sub-function and the interval (domain) over which it is defined.

2. Determine the Range of Each Sub-function: For each sub-function, determine its range within its specified domain. Consider the type of sub-function (linear, quadratic, exponential, etc.) and use appropriate techniques to find its range. Remember to consider the endpoints of the intervals. Are they included (closed brackets [ ]) or excluded (open brackets ( ))? This will significantly impact the range.

3. Combine the Ranges: Once you've determined the range of each sub-function, combine these ranges to find the overall range of the piecewise function. This might involve simply uniting the individual ranges if they are disjoint, or it might require more careful consideration if there's overlap. Pay close attention to whether any gaps exist in the combined range.

4. Consider Endpoints and Continuity: Carefully check the behavior of the function at the boundaries between different sub-functions. Is the function continuous at these points? If not, does this create a gap in the range? This is crucial, as discontinuities can create gaps in the range that need to be accounted for.

Examples: Illustrating the Analytical Method

Let's work through a few examples to solidify our understanding:

Example 1: A Simple Piecewise Function

Consider the following piecewise function:

f(x) = {
  x + 1,  if x < 0
  x² ,   if x ≥ 0
}

1. Sub-functions and Domains:

   • x + 1 for x < 0
   • x² for x ≥ 0

2. Range of Sub-functions:

   • For x + 1 (x < 0), the range approaches ∞ as x approaches 0 from the left, and the range approaches -∞ as x approaches -∞. Therefore, the range is (-∞, 1). Note that 1 is excluded because x cannot be equal to 0.

   • For x² (x ≥ 0), the range is [0, ∞). The range starts at 0 (when x = 0) and increases without bound as x increases.

3. Combining Ranges: Combining the ranges (-∞, 1) and [0, ∞) gives us the overall range of (-∞, ∞) or ℝ (all real numbers). There are no gaps because the two ranges overlap at 0.

Example 2: A Piecewise Function with a Gap

Consider this piecewise function:

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

1. Sub-functions and Domains:

   • -x for x ≤ 1
   • 2x - 3 for x > 1

2. Range of Sub-functions:

   • For -x (x ≤ 1), the range is [-1, ∞).

   • For 2x - 3 (x > 1), the range is (-1, ∞).

3. Combining Ranges: Combining [-1, ∞) and (-1, ∞) gives us the overall range of [-1, ∞). Note that there is no gap, even though the function is discontinuous at x = 1.

Example 3: A More Complex Piecewise Function

Let's consider a more complex example involving a rational function:

h(x) = {
  x²,       if x ≤ -1
  1/(x+2), if -1 < x < 1
  √x,       if x ≥ 1
}

1. Sub-functions and Domains:

   • x² for x ≤ -1
   • 1/(x+2) for -1 < x < 1
   • √x for x ≥ 1

2. Range of Sub-functions:

   • For x² (x ≤ -1), the range is [1, ∞).

   • For 1/(x+2) (-1 < x < 1), let's analyze the behavior: As x approaches -2 from the right, the function approaches -∞. As x approaches -1 from the right, the function approaches 1. As x approaches 1 from the left, the function approaches 1/3. Thus, the range is (-∞, 1) U (1/3, ∞).

   • For √x (x ≥ 1), the range is [1, ∞).

3. Combining Ranges: Combining [1, ∞), (-∞, 1) U (1/3, ∞), and [1, ∞) yields the overall range of (-∞, 1) U [1, ∞). Note the gap caused by the discontinuity in the middle subfunction.

The Graphical Method: A Visual Approach

The graphical method involves plotting the piecewise function. By visually inspecting the graph, you can directly identify the set of y-values covered. This is particularly useful when dealing with more complex functions where the analytical approach might be tedious. Graphing calculators or software like Desmos can be immensely helpful in this process.

Conclusion

Determining the range of a piecewise function requires a careful and systematic approach. Whether you choose the analytical or graphical method, the key is to meticulously examine each sub-function and its domain, accounting for any discontinuities and carefully combining the individual ranges. Understanding the behavior of each sub-function and paying close attention to the endpoints of the intervals are critical steps in achieving an accurate result. By mastering these techniques, you’ll gain a comprehensive understanding of piecewise functions and their properties. Remember to always verify your findings, perhaps by checking a few points within the range to ensure your solution is consistent with the function’s definition.