y = 1 / ((x+1)(x-1))
Find the asymptotes.
Specify what values y can be. (How to find the range of the function?)
Help is appreciated. I already know the answer from desmos calculator but I'd like to know how to find the asymptotes and find the range of the function without desmos.
y = 1 / [ (x + 1) (x - 1) ] = 1 / ( x^2 - 1 )
The vertical asymptotes are the x values that make the denominator = 0
These are the lines x = -1 and x = 1
Since we have a lower degree polynomial over a higher degree polynomial...we will have a horizontal asymptote of 0 [ this will always be true in a lower / higher situation ]
The range is a little difficult
Note that when x is 0, the function will reach its maximum negative value of -1
When x is either a large negative or positive value 1 / [ x^2 - 1 ] will be almost 0
As x moves closer to the left vertical asymptote x = -1 from the left side or closer to the right vertical asymptote x = 1 from the right side...the function aprroaches infinity
For example.....let x = -1.0001 or 1.0001 and we get the large positive value of ≈ 5000
And this will increase as we get closer to each of these asymptotes in this manner
As the function moves closer to the left asymptote of x = -1 form the right or closer to the right vertical asymptote of x = 1 from the left, the function approaches negative infinity
For example when x = -.9999 or x =.9999 the function value ≈ -5000
And this will decrease as we get closer to these aymptotes in this manner
So....the range is ( -infinity , 1 ] U (0 , infinity )
Thank you for your answer.
Did you use any graphic calculator to find the answers or did you figure it out any other way? :)
Thanks again, it's really helpful.