Thread: prove about functions

1) Is this statement true or false.
A rational function can never cross one of its asymptotes?
Justify your answer.

2) What can be said about the sum or difference of
a)two even functions
b)two odd functions
c)an even and odd function

Hello, kishan!

1) Is this statement true or false?
A rational function can never cross one of its asymptotes.
Justify your answer.

A function cannot cross one of its vertical asymptotes.
. . However, it can cross its horizontal asymptote.

Recall that a horizontal asymptote describes the behavior of the graph as $\displaystyle x \to \pm\infty$
That is, the graph may approach a finite value at the extreme right or extreme left;
. . but anything can happen "locally".


Perhaps the simplest example is a Serpentine Curve: .$\displaystyle y \:=\:\frac{x}{x^2+1}$

Code:

                          |
                          |       *
                          |   *       *
                          | *           *
                          |*                *
                          |                       * 
- - - - - - - - - - - - - * - - - - - - - - - - - - -
  *                       |
        *                *|
            *           * |
              *       *   |
                  *       |
                          |

It has a horizontal asymptote: .$\displaystyle y = 0$ .(the x-axis)

But it crosses the x-axis at the origin.

Originally Posted by kishan

2) What can be said about the sum or difference of
a)two even functions
b)two odd functions
c)an even and odd function

a) The sum of two even functions is also an even function.
Let $\displaystyle f,g$ be two even functions.
That means $\displaystyle f(-x)=f(x),g(-x)=g(x),\forall x$
$\displaystyle (f+g)(-x)=f(-x)+g(-x)=f(x)+g(x)=(f+g)(x)$

b) Prove in the same way that the sum of two odd functions is an odd function.

c) The sum of an even function and an odd function can be any kind of function (even, odd, not even or not odd).
For example, $\displaystyle f(x)=x$ is odd, $\displaystyle g(x)=x^2$ is even, but $\displaystyle f(x)+g(x)=x+x^2$ is neither even or odd.