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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1:13 minutes

Problem 9

Textbook Question

Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14.
As x -> -3^-, f(x) -> __

Verified step by step guidance

Identify the behavior of the function as x approaches -3 from the left (x -> -3^-).

Observe the graph near x = -3 to determine the direction in which the function is heading.

Notice that as x approaches -3 from the left, the function f(x) is decreasing rapidly.

Conclude that f(x) is approaching negative infinity as x approaches -3 from the left.

Thus, as x -> -3^-, f(x) -> -∞.

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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 graph has vertical asymptotes at x=6 and x=14, and a horizontal asymptote at y=0.

Limits

Limits describe the behavior of a function as the input approaches a certain value. In this context, evaluating the limit of f(x) as x approaches -3 from the left (denoted as x -> -3^-) helps determine the corresponding output value. Understanding limits is essential for analyzing the continuity and behavior of rational functions near their asymptotes.