A rational function true false problem

Algebra Level 2

True or false :

All rational functions have some form of asymptote, whether it's slant, vertical, or horizontal.

True False

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

4 solutions

Andrew Ellinor
Feb 15, 2016

The rational function f ( x ) = x 3 + x x f(x) = \dfrac{x^3 + x}{x} simplifies to f ( x ) = x 2 + 1 f(x) = x^2 + 1 with a hole at the coordinate ( 0 , 1 ) , (0, 1), but it has no asymptotes!

Proof by counterexample!

Raivat Shah - 3 years ago
Otto Bretscher
Feb 4, 2016

x 2 1 \frac{x^2}{1} has no asymptote.

I'm pretty sure there has to be a variable in both the numerator and the denominator...

Skye Chen - 5 years, 3 months ago

Log in to reply

No, constants are fine. For one thing, we want polynomials to be special cases of rational functions.

Otto Bretscher - 5 years, 3 months ago

Log in to reply

I thought a legitimate rational function must contain at least one unknown in the denominator (but not necessarily a numerator). For x 2 1 \frac{x^{2}}{1} , it can be reduced to x 2 x^{2} and it is equivalent to a quadratic function.

Margaret Zheng - 5 years, 3 months ago

Log in to reply

@Margaret Zheng No, there is no such restriction in the definition of a rational function. Look up the definition from any "legitimate" source

Otto Bretscher - 5 years, 3 months ago

Log in to reply

@Otto Bretscher Ok and I actually just looked it up today. I apologize for the previous comment as it contains false information. Thank you!

Margaret Zheng - 5 years, 3 months ago

Log in to reply

@Margaret Zheng No problem... we are all here to learn (and have fun)

Otto Bretscher - 5 years, 3 months ago

Log in to reply

@Otto Bretscher that's so true and I am aiming to learn a new section of a chapter every single day on Brilliant.org now!

Margaret Zheng - 5 years, 3 months ago
Loo Soo Yong
Jun 27, 2016

You can have curved asymptotes, too

For example the function x^2 +1/(x-1) will have a curved asymptote at y=x^2 when x is infinitely large.

Devin Ky
Feb 18, 2016

All rational functions are in the form p(x)/q(x), where p(x) and q(x) are polynomials. No asymptotes will be present if q(x) has degree 0.

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...