The end behavior of a function is the value to which $f(x)$ tends as $x$ extends to the left or the right infinitely. If $f(x)$ keeps increasing without bound, it is said to tend to positive infinity. The end behavior of this case is stated as up. Conversely, if $f(x)$ keeps decreasing without bound, it is said to tend to negative infinity. In this case, the end behavior is stated as down. For example, consider the graph of a function $g(x).$ From the arrows on the graph, it can be seen that the left end of the graph extends downward, while the right end extends upward. The end behavior of $g$ can then be expressed as follows. To state the end behavior of a function in words, begin by stating the left-end behavior, then state the right-end behavior. A dash can also be used to separate the words. For instance, the end behavior of the graph of $g(x)$ can be written as down and up or as down-up.

End Behavior of Polynomial Functions

When the function is a polynomial function, the end behavior can be determined from the function rule. Particularly, the end behavior is given by the sign of the leading coefficient and the degree of the polynomial. Note that when the degree of the polynomial is even, both ends have the same behavior, which depends on the sign of the leading coefficient. By contrast, when the degree is odd, both ends have opposite behaviors.