Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

5. Rational Functions

Graphing Rational Functions

College Algebra

5. Rational Functions

Graphing Rational Functions

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1:10 minutes

Problem 19

Textbook Question

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

Verified step by step guidance

Identify the horizontal asymptote from the graph, which is the line that the function approaches as x approaches infinity.

Observe that the horizontal asymptote is given by the equation y = 17.

As x approaches infinity, the function f(x) will approach the value of the horizontal asymptote.

Therefore, as x -> ∞, f(x) -> 17.

Conclude that the behavior of the function as x approaches infinity is determined by the horizontal asymptote.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.