To find whether the statement “ The graph of a rational function can never cross one of its asymptotes” is true or false.
Answer to Problem 94E
False
Explanation of Solution
Given:
Statement: The graph of a rational function can never cross one of its asymptotes.
The graph of a rational function can cross one of its asymptotes.
Example:
Consider a rational function,
Calculation:
Asymptotes:
Vertical asymptotes:
To find vertical asymptotes, put denominator of the given function equal to zero.
Horizontal asymptotes:
As the degree of numerator is equal to the degree of the denominator, the horizontal asymptote is
Now draw the graph of the function.
Calculation for graph:
Consider
Values of x | Values of f (x) |
0 | -0.6 |
1 | Infinity |
-1 | -0.75 |
2 | 2.142 |
-2 | -1.666 |
By taking different values of x, the graph can be plotted.
Graph:
Interpretation:
From the above graph, it is clear that, the curve of rational function crosses its horizontal asymptote y = 3.
So, a rational function can cross one of its asymptotes.
Conclusion:
Therefore, the given statement is false.
Chapter 2 Solutions
Precalculus with Limits: A Graphing Approach
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning