End Behavior Of A Function Rules

The end behavior of a function describes what happens to the function's output (y-values) as the input (x-values) grow without bound, either positively (approaching positive infinity) or negatively (approaching negative infinity). Understanding end behavior provides crucial information about the overall shape and characteristics of a function, especially for polynomial and rational functions, and is a fundamental concept in calculus and analysis.

Understanding End Behavior

The end behavior of a function is essentially the trend the function follows as x gets extremely large (positive infinity, denoted as ∞) or extremely small (negative infinity, denoted as -∞). We are not concerned with the function's behavior at specific points or even over a finite interval, but rather its ultimate direction. This is typically described using limit notation, where we express the limit of the function as x approaches positive or negative infinity.

Formally, we investigate two limits:

1. Limit as x approaches positive infinity: lim x→∞ f(x)
2. Limit as x approaches negative infinity: lim x→-∞ f(x)

The results of these limits can be a finite number, positive infinity, negative infinity, or the limit may not exist (oscillating behavior). For many common functions, especially polynomial and rational functions, the end behavior is relatively straightforward to determine based on specific rules.

Rules for Determining End Behavior

The rules for determining the end behavior of a function vary depending on the type of function. We will primarily focus on polynomial and rational functions, as they are commonly encountered and have well-defined rules.

1. Polynomial Functions

A polynomial function is defined as:

f(x) = anx**n + a**n-1x**n-1 + ... + a1x + a0

where a**n, a**n-1, ..., a1, a0 are constants (coefficients) and n is a non-negative integer (the degree of the polynomial).

The end behavior of a polynomial function is determined solely by its leading term, which is the term with the highest power of x (anx**n). The coefficient a**n is called the leading coefficient.

Here's how the leading term dictates the end behavior:

• Even Degree (n is even):

  • If a**n > 0 (positive leading coefficient): As x approaches both positive and negative infinity, f(x) approaches positive infinity. In limit notation:

    lim x→∞ f(x) = ∞ and lim x→-∞ f(x) = ∞

  • If a**n < 0 (negative leading coefficient): As x approaches both positive and negative infinity, f(x) approaches negative infinity. In limit notation:

    lim x→∞ f(x) = -∞ and lim x→-∞ f(x) = -∞

  In simpler terms, even degree polynomials with a positive leading coefficient open upwards, resembling a "U" shape on the far left and right. Even degree polynomials with a negative leading coefficient open downwards, resembling an upside-down "U" shape.

• Odd Degree (n is odd):

  • If a**n > 0 (positive leading coefficient): As x approaches positive infinity, f(x) approaches positive infinity. As x approaches negative infinity, f(x) approaches negative infinity. In limit notation:

    lim x→∞ f(x) = ∞ and lim x→-∞ f(x) = -∞

  • If a**n < 0 (negative leading coefficient): As x approaches positive infinity, f(x) approaches negative infinity. As x approaches negative infinity, f(x) approaches positive infinity. In limit notation:

    lim x→∞ f(x) = -∞ and lim x→-∞ f(x) = ∞

  Odd degree polynomials with a positive leading coefficient rise to the right and fall to the left. Odd degree polynomials with a negative leading coefficient fall to the right and rise to the left.

Examples of Polynomial End Behavior:

1. f(x) = 3x4 - 2x2 + x - 5

   • Leading term: 3x4
   • Degree: 4 (even)
   • Leading coefficient: 3 (positive)
   • End behavior: As x → ∞, f(x) → ∞ and as x → -∞, f(x) → ∞

2. g(x) = -x5 + 4x3 + 2x

   • Leading term: -x5
   • Degree: 5 (odd)
   • Leading coefficient: -1 (negative)
   • End behavior: As x → ∞, g(x) → -∞ and as x → -∞, g(x) → ∞

3. h(x) = -2x6 + x3 - 1

   • Leading term: -2x6
   • Degree: 6 (even)
   • Leading coefficient: -2 (negative)
   • End behavior: As x → ∞, h(x) → -∞ and as x → -∞, h(x) → -∞

4. k(x) = 5x3 - 7x + 2

   • Leading term: 5x3
   • Degree: 3 (odd)
   • Leading coefficient: 5 (positive)
   • End behavior: As x → ∞, k(x) → ∞ and as x → -∞, k(x) → -∞

2. Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomial functions:

f(x) = p(x) / q(x)

where p(x) and q(x) are polynomial functions. The end behavior of a rational function is determined by comparing the degrees of the numerator and denominator polynomials.

Let n be the degree of p(x) and m be the degree of q(x).

• Case 1: n < m (Degree of numerator is less than the degree of the denominator)

  In this case, as x approaches positive or negative infinity, the function f(x) approaches 0. The x-axis (y = 0) is a horizontal asymptote.

  lim x→∞ f(x) = 0 and lim x→-∞ f(x) = 0

  Example: f(x) = (x + 1) / (x2 + 2x* + 1). The degree of the numerator is 1 and the degree of the denominator is 2. Therefore, the end behavior is that the function approaches 0 as x goes to positive or negative infinity.*

• Case 2: n = m (Degree of numerator is equal to the degree of the denominator)

  In this case, as x approaches positive or negative infinity, the function f(x) approaches the ratio of the leading coefficients of the numerator and denominator polynomials. If the leading coefficient of p(x) is a**n and the leading coefficient of q(x) is b**m, then the function approaches a**n / b**m. The line y = a**n / b**m is a horizontal asymptote.

  lim x→∞ f(x) = a**n / b**m and lim x→-∞ f(x) = a**n / b**m

  Example: f(x) = (3x2 + 2x* + 1) / (2x2 - x + 5). The degree of both the numerator and denominator is 2. The leading coefficient of the numerator is 3 and the leading coefficient of the denominator is 2. Therefore, the end behavior is that the function approaches 3/2 as x goes to positive or negative infinity.*

• Case 3: n > m (Degree of numerator is greater than the degree of the denominator)

  In this case, the end behavior is more complex. The function does not approach a horizontal asymptote. Instead, it either approaches positive infinity, negative infinity, or follows a slant (oblique) asymptote. To determine the precise end behavior, we need to consider the specific polynomials.

  • If n = m + 1, there is a slant asymptote. We can find the equation of the slant asymptote by performing polynomial long division. The quotient obtained from the division (ignoring the remainder) is the equation of the slant asymptote. The end behavior then follows the behavior of this slant asymptote.
  • If n > m + 1, there is no slant asymptote. The function will approach either positive or negative infinity, depending on the signs of the leading coefficients and the parity (even or odd) of the difference n - m.

  Example 1 (Slant Asymptote): f(x) = (x2 + 1) / (x). The degree of the numerator is 2 and the degree of the denominator is 1. Performing polynomial long division, we get x + (1/x). As x approaches infinity, 1/x approaches 0, so the slant asymptote is y = x. Therefore, as x → ∞, f(x) → ∞ and as x → -∞, f(x) → -∞.

  Example 2 (No Slant Asymptote): f(x) = (x3 + 1) / (x). The degree of the numerator is 3 and the degree of the denominator is 1. The difference is 2, so there's no slant asymptote. As x → ∞, f(x) → ∞ and as x → -∞, f(x) → ∞.

Examples of Rational Function End Behavior:

1. f(x) = (2x + 1) / (x2 - 3)

   • Degree of numerator: 1
   • Degree of denominator: 2
   • n < m: Therefore, the end behavior is lim x→∞ f(x) = 0 and lim x→-∞ f(x) = 0

2. g(x) = (4x2 - 5) / (2x2 + x)

   • Degree of numerator: 2
   • Degree of denominator: 2
   • n = m: Leading coefficient of numerator = 4, leading coefficient of denominator = 2. Therefore, the end behavior is lim x→∞ g(x) = 4/2 = 2 and lim x→-∞ g(x) = 2

3. h(x) = (x3 + 2x) / (x - 1)

   • Degree of numerator: 3
   • Degree of denominator: 1
   • n > m: The function does not have a horizontal asymptote. As x → ∞, h(x) → ∞ and as x → -∞, h(x) → ∞. There is no slant asymptote because the difference in degrees is greater than 1.

4. k(x) = (x2 + 3x - 2) / (x + 1)

   • Degree of numerator: 2
   • Degree of denominator: 1
   • n > m: The function does not have a horizontal asymptote. Since n = m + 1, there is a slant asymptote. Performing polynomial long division gives us x + 2 - (4/x + 1). So the slant asymptote is y = x + 2. As x → ∞, k(x) → ∞ and as x → -∞, k(x) → -∞.

3. Exponential Functions

Exponential functions have the general form f(x) = a**x, where a is a constant called the base and a > 0, a ≠ 1.

• If a > 1:
  • As x approaches positive infinity, f(x) approaches positive infinity.
  • As x approaches negative infinity, f(x) approaches 0. The x-axis is a horizontal asymptote on the left.
• If 0 < a < 1:
  • As x approaches positive infinity, f(x) approaches 0. The x-axis is a horizontal asymptote on the right.
  • As x approaches negative infinity, f(x) approaches positive infinity.

Examples of Exponential Function End Behavior:

1. f(x) = 2x

   • Base a = 2 > 1
   • As x → ∞, f(x) → ∞
   • As x → -∞, f(x) → 0

2. g(x) = (1/3)x

   • Base a = 1/3, 0 < a < 1
   • As x → ∞, g(x) → 0
   • As x → -∞, g(x) → ∞

4. Other Functions

• Logarithmic Functions: Logarithmic functions, such as f(x) = log(x), only have a domain for x > 0. Therefore, we only consider the end behavior as x approaches positive infinity. As x approaches positive infinity, log(x) approaches positive infinity (though very slowly).

• Trigonometric Functions: Trigonometric functions like sine and cosine oscillate between -1 and 1 and do not approach any specific value as x approaches positive or negative infinity. Therefore, the limit does not exist.

Importance of Understanding End Behavior

Understanding the end behavior of a function is vital for several reasons:

• Graphing: Knowing the end behavior helps in sketching the graph of a function. It provides a framework for understanding how the function behaves for large values of x.
• Analysis: In calculus, end behavior is essential for determining limits at infinity, identifying horizontal and slant asymptotes, and analyzing the convergence or divergence of integrals.
• Modeling: When using functions to model real-world phenomena, the end behavior provides insights into long-term trends and predictions. For example, in population growth models, the end behavior might indicate whether a population will stabilize, grow indefinitely, or decline to extinction.
• Comparison of Functions: End behavior allows us to compare the growth rates of different functions. For instance, an exponential function will eventually grow faster than any polynomial function, regardless of the polynomial's degree.
• Root Finding: While end behavior alone doesn't pinpoint roots, it helps understand the number of potential roots and their possible locations based on the function's overall trend.
• Asymptotes: End behavior directly informs the presence and nature of horizontal and slant asymptotes, critical features of rational functions.

Limitations and Considerations

While the rules outlined above are effective for many common functions, there are situations where determining the end behavior is more complex:

• Piecewise Functions: The end behavior of a piecewise function must be evaluated separately for each piece that extends to infinity.
• Transcendental Functions: Functions involving trigonometric, exponential, logarithmic, and other transcendental elements can have more complex end behaviors that might require more advanced techniques to analyze.
• Oscillating Functions: Functions that oscillate indefinitely as x approaches infinity may not have a well-defined limit, and their end behavior might be described as "oscillating" or "non-existent."
• Functions with Radicals: Functions containing radical expressions (square roots, cube roots, etc.) can sometimes require careful algebraic manipulation to determine the end behavior. Often multiplying by the conjugate is required.

Practical Examples

Let's consider some practical examples of how understanding end behavior can be useful:

1. Analyzing the Profit Function of a Company: Suppose a company's profit (in millions of dollars) is modeled by the function P(x) = -0.1x4 + x3 - 2x2 + 5x, where x represents the number of years since the company's inception. The leading term is -0.1x4. Since the degree is even and the leading coefficient is negative, the end behavior is that as x approaches infinity, P(x) approaches negative infinity. This indicates that, according to this model, the company's profits will eventually decline to a significant loss in the long run. While not necessarily predictive of the actual future (models have limitations), it raises a red flag.

2. Modeling the Spread of a Disease: A simplified model for the number of infected individuals I(t) in a population t days after the initial outbreak might be given by I(t) = (1000*t) / (t + 5). This is a rational function. The degrees of the numerator and denominator are both 1. The leading coefficient of the numerator is 1000 and the leading coefficient of the denominator is 1. Therefore, as t approaches infinity, I(t) approaches 1000. This suggests that the disease will eventually infect approximately 1000 people, according to this model.

3. Determining the Dosage of a Drug: A drug's concentration in the bloodstream after t hours is modeled by C(t) = 5e**-0.2t. As t approaches infinity, C(t) approaches 0. This means that the concentration of the drug in the bloodstream decreases over time, approaching zero in the long run. This helps to determine when and if another dose is needed.

Conclusion

The end behavior of a function is a fundamental concept in mathematics that provides valuable insights into the long-term trends and overall characteristics of the function. By understanding the rules for determining end behavior, particularly for polynomial and rational functions, we can gain a deeper understanding of the function's graph, its limits at infinity, and its applications in modeling real-world phenomena. While the rules are generally straightforward, it is important to be aware of the limitations and complexities that can arise with more advanced or specialized functions. Mastery of end behavior analysis is a crucial step in developing a strong foundation in calculus and mathematical analysis.