Use the graph of the rational function in the figure shown to complete each statement in Exercises 15–20.
As x -> ∞, f(x) -> __

Key Concepts

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. The general form is f(x) = P(x)/Q(x), where P and Q are polynomials. Understanding rational functions is crucial for analyzing their behavior, including identifying asymptotes and intercepts, which are key to interpreting their graphs.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. Vertical asymptotes occur where the function is undefined, typically where the denominator equals zero, while horizontal asymptotes indicate the behavior of the function as x approaches infinity. In this case, the horizontal asymptote at y=17 suggests that as x approaches infinity, f(x) approaches 17.

End Behavior of Functions

The end behavior of a function describes how the function behaves as the input values approach positive or negative infinity. For rational functions, this behavior is often determined by the degrees of the numerator and denominator polynomials. In this scenario, knowing the horizontal asymptote helps predict that as x approaches infinity, f(x) will approach the value of the horizontal asymptote.