End Behavior of a Function

In mathematics, end behaviour refers to the way a function behaves as the input (often denoted as xxx) becomes very large or very small, either positively or negatively. By analysing the end behaviour of a function, we can understand how the function behaves at the extremes of its domain. This concept is crucial, especially when working with polynomial functions, rational functions, and others that exhibit distinct behaviours as xxx approaches large or small values.

What Is End Behaviour?

The end behaviour of a function describes how the function behaves as the input values (usually represented by xxx) either approach positive infinity or negative infinity. Mathematically, we are interested in the limits of the function.

• As x→∞: What happens to the function as x gets larger and larger?
• As x→−∞: What happens to the function as x gets smaller and smaller?

These behaviours are particularly useful for sketching graphs and analysing the long-term trends of a function.

How to Determine the End Behaviour

There are several ways to determine the end behavior of a function, depending on the type of function you're working with. Let's explore the end behavior of some common types of functions.

1. Polynomial Functions

For polynomial functions, the end behavior is determined primarily by the leading term, which is the term with the highest degree.

Example: f(x)=x³−4x+5

• The leading term is x³, so the behavior of the function as x→∞ or x→−∞ will be similar to the behavior of x³.
• For odd-degree polynomials (like x³), as x→∞, f(x)→∞, and as x→−∞, f(x)→−∞.

Thus, the end behavior for f(x)=x³−4x+5 is:

• As x→∞, f(x)→∞
• As x→−∞, f(x)→−∞

Example: f(x)=−x²+3x−2

• The leading term is −x², so the behavior of the function will resemble that of −x².
• For even-degree polynomials (like x²), as x→∞, f(x)→−∞, and as x→−∞, f(x)→−∞f(x).

2. Rational Functions

For rational functions, the end behavior is determined by the degrees of the numerator and the denominator.

Dependence on Function Type

• The end behavior is determined by the highest degree term in a polynomial function, the leading term in a rational function, or the base of an exponential or logarithmic function.

Trigonometric Functions

• Periodic functions like sine and cosine do not have end behavior in the traditional sense, as they oscillate indefinitely between fixed values.

End Behavior of Common Function Types

1. Polynomial Functions
   The end behavior depends on the degree and leading coefficient of the polynomial.
2. Rational Functions
   The end behavior is determined by the relationship between the degrees of the numerator and the denominator.
3. Exponential Functions
   The end behavior is influenced by the base of the exponential.
4. Logarithmic Functions
   The end behavior depends on the behavior of the logarithmic scale as the input approaches certain values.
5. Trigonometric Functions
   These functions exhibit oscillatory behavior, so traditional end behavior is not defined.

Importance of End Behavior

Understanding end behavior is essential in:

• Sketching graphs of functions.
• Analysing the trends of functions in applied problems.
• Identifying limits at infinity in calculus.