Find The Domain Of The Following Piecewise Function

Navigating the realm of piecewise functions can seem daunting at first, but understanding their domain is a fundamental step in mastering these mathematical entities. The domain of a piecewise function essentially outlines the set of all possible input values (often represented as 'x') for which the function is defined. By systematically analyzing each piece of the function, we can construct a comprehensive view of its overall domain, paving the way for accurate evaluations and graphical representations.

Understanding Piecewise Functions

A piecewise function, as the name suggests, is a function defined by multiple sub-functions, each applying to a certain interval of the main function's domain. These intervals might be continuous, disjoint, or overlapping, contributing to the function's unique characteristics. The key to identifying the domain lies in understanding the intervals over which each piece is active.

• Definition: A piecewise function is defined by different formulas on different parts of its domain.
• Notation: It is typically written using a curly brace notation, where each line specifies a sub-function and the interval over which it applies.
• Example:

  f(x) = {
      x^2,   if x < 0
      x + 1, if 0 <= x <= 2
      3,     if x > 2
  }
  

  In this example, the function behaves like x² when x is less than 0, like x + 1 when x is between 0 and 2 (inclusive), and it is constant at 3 when x is greater than 2.

Identifying the Domain of a Piecewise Function: A Step-by-Step Guide

Finding the domain of a piecewise function involves a careful examination of each piece and its corresponding interval. The process can be broken down into several manageable steps:

Step 1: Examine Each Piece Individually

Begin by looking at each sub-function separately. Identify any domain restrictions inherent to that function. This is crucial because certain functions have limitations.

• Polynomials: Polynomial functions (e.g., x², x + 1) are defined for all real numbers.
• Rational Functions: Rational functions (e.g., 1/x) are undefined where the denominator is zero. You must exclude these values from the domain.
• Radical Functions: Even-indexed radical functions (e.g., √x) are only defined for non-negative numbers. The expression inside the radical must be greater than or equal to zero.
• Logarithmic Functions: Logarithmic functions (e.g., log(x)) are only defined for positive numbers. The argument of the logarithm must be greater than zero.

Step 2: Note the Interval for Each Piece

For each sub-function, carefully note the interval over which it is defined. This is usually given as an inequality next to the function definition. Pay close attention to whether the endpoints of the interval are included (indicated by ≤ or ≥) or excluded (indicated by < or >).

• Closed Interval: An interval that includes its endpoints (e.g., [0, 2]).
• Open Interval: An interval that excludes its endpoints (e.g., (0, 2)).
• Half-Open Interval: An interval that includes one endpoint but excludes the other (e.g., [0, 2) or (0, 2]).

Step 3: Combine the Intervals

Once you have identified the intervals for each piece, combine them to find the overall domain of the piecewise function. This involves checking for any gaps or overlaps between the intervals.

• Overlapping Intervals: If intervals overlap, the function is defined for the combined range of those intervals.
• Disjoint Intervals: If intervals are disjoint (separate), the function is defined on each interval separately. The domain is the union of these intervals.
• Gaps: If there are gaps between intervals, the function is not defined for the values in those gaps. These values must be excluded from the domain.

Step 4: Express the Domain in Interval Notation

Finally, express the domain in interval notation. This is a standard way of representing sets of real numbers using brackets and parentheses.

• Parentheses ( ) indicate that the endpoint is not included.
• Brackets [ ] indicate that the endpoint is included.
• The symbol ∞ represents infinity. It is always enclosed in a parenthesis because infinity is not a specific number and cannot be included.
• The symbol ∪ represents the union of two or more intervals.

Example 1: A Simple Piecewise Function

Let's revisit the example from earlier:

f(x) = {
    x^2,   if x < 0
    x + 1, if 0 <= x <= 2
    3,     if x > 2
}

1. Examine Each Piece:
   • x²: This is a polynomial, defined for all real numbers.
   • x + 1: This is a polynomial, defined for all real numbers.
   • 3: This is a constant function, defined for all real numbers.
2. Note the Interval for Each Piece:
   • x²: x < 0 (Open interval)
   • x + 1: 0 ≤ x ≤ 2 (Closed interval)
   • 3: x > 2 (Open interval)
3. Combine the Intervals:
   • The intervals are (-∞, 0), [0, 2], and (2, ∞).
   • Notice that the interval [0, 2] includes 0, which is the endpoint of the first interval. Similarly, the interval (2, ∞) starts immediately after 2, which is the endpoint of the second interval.
   • There are no gaps or overlaps. The intervals connect seamlessly.
4. Express the Domain in Interval Notation:
   • The domain is (-∞, 0) ∪ [0, 2] ∪ (2, ∞).
   • This simplifies to (-∞, ∞), which means the domain is all real numbers.

Example 2: A Piecewise Function with a Restriction

Consider the following piecewise function:

g(x) = {
    1/x,   if x < -1
    x,     if -1 <= x <= 1
    √(x-1), if x > 1
}

1. Examine Each Piece:
   • 1/x: This is a rational function. It is undefined when x = 0.
   • x: This is a polynomial, defined for all real numbers.
   • √(x-1): This is a radical function. It is only defined when x - 1 ≥ 0, which means x ≥ 1.
2. Note the Interval for Each Piece:
   • 1/x: x < -1 (Open interval)
   • x: -1 ≤ x ≤ 1 (Closed interval)
   • √(x-1): x > 1 (Open interval)
3. Combine the Intervals:
   • The intervals are (-∞, -1), [-1, 1], and (1, ∞).
   • The function 1/x has a restriction at x = 0, but this is not within its interval x < -1, so it doesn't affect the overall domain.
   • The function √(x-1) has a restriction of x ≥ 1. Its interval is x > 1, which is consistent with its restriction. However, because the interval is x > 1 (and not x ≥ 1), the value x = 1 is not included in this piece's definition.
   • The intervals connect seamlessly, but we need to consider whether x = 1 is actually in the domain. The second piece includes x = 1, and the third piece excludes it. Therefore, x = 1 is in the domain.
4. Express the Domain in Interval Notation:
   • The domain is (-∞, -1) ∪ [-1, 1] ∪ (1, ∞).
   • This simplifies to (-∞, ∞), which means the domain is all real numbers.

Example 3: A Piecewise Function with a Gap

Let's analyze this function:

h(x) = {
    x + 2, if x < 0
    x - 2, if x > 0
}

1. Examine Each Piece:
   • x + 2: This is a polynomial, defined for all real numbers.
   • x - 2: This is a polynomial, defined for all real numbers.
2. Note the Interval for Each Piece:
   • x + 2: x < 0 (Open interval)
   • x - 2: x > 0 (Open interval)
3. Combine the Intervals:
   • The intervals are (-∞, 0) and (0, ∞).
   • Notice that x = 0 is not included in either interval. There is a gap at x = 0.
4. Express the Domain in Interval Notation:
   • The domain is (-∞, 0) ∪ (0, ∞).
   • This means the domain is all real numbers except for 0.

Common Mistakes and How to Avoid Them

Finding the domain of piecewise functions can be tricky, and it's easy to make mistakes. Here are some common errors to watch out for:

• Ignoring Restrictions: Forgetting to consider the domain restrictions of individual sub-functions (e.g., rational, radical, or logarithmic functions).
• Misinterpreting Intervals: Failing to correctly interpret the inequalities that define the intervals, especially whether endpoints are included or excluded.
• Overlooking Gaps: Missing gaps in the domain where the function is not defined.
• Incorrect Notation: Using incorrect interval notation to express the domain.

Tips to Avoid Mistakes:

• Write it Out: Clearly write down the domain of each piece separately before combining them.
• Number Line Visualization: Use a number line to visualize the intervals and identify any gaps or overlaps.
• Double-Check Endpoints: Carefully check whether the endpoints of each interval are included or excluded.
• Consider Restrictions First: Identify any domain restrictions for each sub-function before considering the intervals.

The Importance of Domain

Understanding the domain of a piecewise function is not just an academic exercise. It is essential for several reasons:

• Accurate Evaluation: The domain tells you which input values are valid for the function. Trying to evaluate the function outside its domain will lead to undefined results.
• Correct Graphing: The domain determines the extent of the graph of the function. You should only plot the function for values within its domain.
• Valid Modeling: When using piecewise functions to model real-world phenomena, the domain represents the range of values for which the model is applicable.

Advanced Examples

Let's explore some more complex examples to solidify your understanding.

Example 4: A Piecewise Function with Nested Restrictions

f(x) = {
    √(x+4),        if -4 <= x < 0
    x^2 / (x-2),   if 0 <= x <= 4
    log(x-3),      if x > 4
}

1. Examine Each Piece:
   • √(x+4): Defined for x + 4 ≥ 0, which means x ≥ -4.
   • x² / (x-2): Defined for all x except x = 2.
   • log(x-3): Defined for x - 3 > 0, which means x > 3.
2. Note the Interval for Each Piece:
   • √(x+4): -4 ≤ x < 0 (Half-open interval)
   • x² / (x-2): 0 ≤ x ≤ 4 (Closed interval)
   • log(x-3): x > 4 (Open interval)
3. Combine the Intervals:
   • √(x+4): The function is defined for x ≥ -4, and the interval is -4 ≤ x < 0. This piece is valid over the entire interval.
   • x² / (x-2): The function is undefined at x = 2. The interval is 0 ≤ x ≤ 4. Therefore, we must exclude x = 2 from this interval.
   • log(x-3): The function is defined for x > 3, and the interval is x > 4. This piece is valid over the entire interval.
4. Express the Domain in Interval Notation:
   • The first interval is [-4, 0).
   • The second interval is [0, 2) ∪ (2, 4]. We exclude x = 2.
   • The third interval is (4, ∞).
   • Combining these, we have: [-4, 0) ∪ [0, 2) ∪ (2, 4] ∪ (4, ∞).
   • Since [-4,0) ∪ [0,2) = [-4,2), the domain can be simplified to: [-4, 2) ∪ (2, 4] ∪ (4, ∞)

Example 5: A More Abstract Example

g(x) = {
    f_1(x), if x ∈ A
    f_2(x), if x ∈ B
}

Here, f₁(x) and f₂(x) are arbitrary functions, and A and B are sets of real numbers. The domain of g(x) is simply the union of the sets A and B (A ∪ B), provided that f₁(x) is well-defined on A and f₂(x) is well-defined on B. This example emphasizes the general principle that the domain of a piecewise function is constructed by piecing together the domains of its constituent sub-functions according to the intervals they are defined on.

Conclusion

Finding the domain of a piecewise function is a vital skill for anyone working with these versatile mathematical tools. By carefully examining each piece, noting the intervals over which they apply, and considering any domain restrictions, you can accurately determine the overall domain of the function. Remember to express the domain in interval notation and be mindful of common mistakes. With practice, you'll become proficient in unraveling the domains of even the most complex piecewise functions. Understanding the domain enables accurate function evaluation, proper graphing, and valid modeling, solidifying its importance in various mathematical and real-world applications.