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

We know that `x^2 ge AA x in R.`
`rArr x^2+2 ge 2, AA x in R`
or `2 le (x^2 +2) gt oo`
or `1/2 ge (1)/(x^2 + 2) gt 0 `
`rArr 0 lt (1)/(x^2 +2) lt 1/2 `
(ii)`1/(x^2-2x + 3) = (1) /( x-1)^2+ 2` ltbr gt Now we know that `(x-1)^2 ge o AA x in R.` Thus
`(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 )^2ge 0, AA in R`
`rArr ( x -1/2)^2-5/4ge -5/4, AA x in R`
For `(1)/((x-1/2)^2-5/4)`, we must have
`(x - 1/2)^2- 5/4 in [-5/40) cup (0,oo)`
`rArr (1)/((x- 1/2)^2- 5/4 )in(- oo , -4/5]cup(0,oo)`