Question

How do you find the domain and range of `f(x) = 2 / (1 - x²)`?

Answer 1

Domain: `{x|x in RR, x!=1, x!=-1}`

Range: `{f(x)|f(x) in RR, x<0 or x>=2}`

Explanation:

`f(x)=2/(1-x^2)`

To find the Domain find where the function is undefined, i.e. `a/0`

To do this we set the denominator equal to zero and solve, notice it is a difference of squares:

`1-x^2=0`

`(1-x)(1+x)=0`

`x=1` and `x=-1` so the domain is all real numbers except those two:

`{x|x in RR, x!=1, x!=-1}`

Now the range, as `x` gets really big positive or negative the `1` becomes insignificant so we must only consider `1/-x`, sense we know as

`2/-x ->+-oo, f(x) -> 0` then:

`2/(1-x^2) ->+-oo, f(x) -> 0`

so we know for numbers above and below the asymptotes the range is `-oo " to " 0`, what about between the asymptotes? In that case the numbers input are `-1<=x<=1` so at 0 the range is 2 and the closer we get to -1 or 1 the range is `2/"some tiny number" -> oo`, so that range is `2 " to " oo`

Putting it all together:

`{f(x)|f(x) in RR, x<0 or x>=2}`

graph{2/(1-x^2) [-10, 10, -5, 5]}