Finding the asymptotes of an odd function

Question

The question gave the graph shown below, giving one asymptote as $y=f(x)=\frac{1}{2}x - 3$. Graph:

I was asked to complete the rest of the graph and draw on any missing asymptotes, and this is what I drew:

For the other asymptote which I drew I wasn't sure what its equation was but I predict it's $y=\frac{1}{2}x + 3$. My question is how would I know for sure that that is the equation of the second asymptote without just guessing?

Answer 1

the given asymptote means

$$\lim_{x\to -\infty}(f (x)-(\frac {1}{2}x-3))=0$$

but $f $ is odd $ (f (-x)=-f (x)) $.

replacing $x $ by $-x $, we get

$$\lim_{x\to+\infty}(f (-x)-(\frac {-1}{2}x-3))=0$$

$$\implies$$ $$\lim_{+\infty}(f (x)-(\frac {1}{2}x+3))=0$$ thus the other asymptote is

$$y=\frac {1}{2}x+3$$

Answer 2

$$ y= x/2 +3$$

for an odd function

$$ x\rightarrow -x , \, y\rightarrow -y $$

$$ -y= -x/2 +3$$

or

$$ y= x/2 -3$$