Find the domain and range of the function `f={(x :1/(1-x^2)): x in R , x!=+-1}dot`

Explanation

We have, `f(x)=(1)/((1-x^(2))),x""inR`.
Clearly, f(x) is defined for all real values of x for which `(1-x^(2))ne0`.
Now, `(1-x^(2))=0implies(1+x)(1-x)(1-x)=0impliesx=-1orx=+1`.
Thus, f(x) is defined for all values of `x""inR` except `pm1`.
`:."dom "(f)=R-{-1,1}`.
Let y=f(x). Then,
`y=(1)/(1-x^(2))impliesy-x^(2)y=1impliesx^(2)y=y-1`
`impliesx^(2)=(y-1)/(y)impliesx=pmsqrt((y-1)/(y))." "......(i)`
It is clear from (i) that x will take real values only when `(y-1)/(y)ge0`.
Now, `(y-1)/(y)ge0iff(y-1le0andylt0)or(y-1ge0andygt0)`
`iff(yle1andylt0)or(yge1andygt0)`
`iff(ylt0)or(yge1)`
`iffyin(-oo,0)or[1,oo)`.
`:."range "(f)=(-oo,0)uu[1,oo)`.
Hence, dom `(f)=R-{-1,1}"and range "(f)=(-oo,0)uu[1,oo)`.