how to identify the vertical, horizontal, domain, and range of each asymptotes

Question

 g(x)=6/(x-3)

Answer 1

Hello l'Onie,

 

Let's start with the parent function f(x) = 1/x.

 

The Vertical Asymptote is going to be x = 0, since this is what is going to make the denominator zero.

 

The Horizontal Asymptote is going to be y = 0.  Why?

Well, the limit as x→∞ = 0, since as the denominator gets really large, the overall fraction, 1/x, will get really small.

And the limit as x→-∞ = 0, since as the denominator gets really large (negative-wise), the overall fraction, 1/x, will again get really small.

 

For the domain, these are the x-values that you can use, so all real numbers except zero:

Domain: (-∞,0) ∪ (0,∞)

 

For the range, since you have a horizontal asymptote of y=0, which the graph will never cross in this case, the range will be the same:

Range: (-∞,0) ∪ (0,∞)

 

Now for the given function: g(x)=6/(x-3).  Multiplying 1/x by 6 won't affect any of the above, but the '-3' in the denominator means we would need a +3 to make the denominator zero, so this will move the Vertical Asymptote 3 units to the right.  So the domain is now:

Domain: (-∞,3) ∪ (3,∞) 

Vertical Asymptote: x = 3

 

No movement/transformation is being made up or down, so the range will remain the same

Range: (-∞,0) ∪ (0,∞) 

 

Again, since no movement up or down of the parent function, the Horizontal Asymptote stays put:

y=0

 

Hope this helps.

Michael Ehlers

Answer 2

Hi l'Onie

 

Let's look at vertical asymptotes first and domain first.

To do this we focus in on the denominator which is x-3

Set this equal to zero and solve for x

x-3=0

x=3

 

So the domain is (-∞,3) u (3,∞)

This can also be written as all Reals such that x≠3

 

The vertical asymptote is also x=3

 

For the horizontal asymptote, we look at the degree of the numerator and denominator.  In this case there is no variable in the numerator so the degree is 0.  The denominator (x-3) is degree 1.  It's the exponent on the variable x.  Since 0<1, then the horizontal asymptote is the x-axis or y=0

 

Finally, the range.

Since the horizontal asymptote is y=0, the range is (-∞,0) u (0,∞)
This can also be written as all Reals such that y≠0