Find all the possible values of the following expressions: `1/(x^2+2)` (ii) `1/(x^2-2x+3)` (iii) `1/(x^2-x-1)`

Explanation

(i) WE know that `x^(2) ge 0 AA x in R.`
`implies x^(2)+2 ge 2, AA x in R.`
or `2 le (x^(2)+2) lt oo`
or ` (1)/(2) ge (1)/(x^(2)+2) gt 0`
`implies 0 lt (1)/(x^(2)+2) le (1)/(2)`
(ii) `(1)/(x^(2)-2x+3)=(1)/((x-1)^(2) +2)`
Now we know that `(x-1)^(2) ge 0 AA x in R.` Thus,
`(x-1)^(2) +2 ge 2 AA x in R. `
or ` 2 le (x-1)^(2)+2 lt oo`
or `(1)/(2) ge (1)/((x-1)^(2) +2) gt 0`
or ` (1)/(x^(2)-2x+3) in (0,(1)/(2)]`
(iii) `(1)/(x^(2)-x-1)=(1)/((x-(1)/(2))^2-(5)/(4))`
`(x-(1)/(2))^(2) ge 0, AA x in R`
`implies(x-(1)/(2))^(2) -(5)/(4) ge -(5)/(4), AA x in R`
For `(1)/((x-(1)/(2))^2-(5)/(4)),` we must have
`(x-(1)/(2))^(2) -(5)/(4) in [-(5)/(4),0] cup (0,oo)`
`implies(1)/((x-(1)/(2))^2-(5)/(4)) in (-oo,-(4)/(5)] cup (0,oo)`