Find the rang of `f(x)=(x^2+34 x-71)/(x^2+2x-7)` `f(x)=(x^2-x+1)/(x^2+x+1)`

Answer

`(a) (-infty, 5 )cup [9, infty)`
`[1//3,3]`

Explanation

Let `(x^(2) 34 x - 71)/(x^(2) + 2x - 7) = y`
or `( 1 - y) + 2(17 - y) x + (7y - 71) = 0`
For real value of x
`b^(2) - 4ac ge 0`
`rArr y^(2) - 14 y + 45 ge 0`
`rArr (y - 5) (y - 9) ge 0`
`rArr y le 5 or y ge 9`
Hence, the range is `(- infty, 5] cup [ 9, infty)`
(b) Let `y= (x^(2) - x + 1)/(x^(2)+ x + 1)`
`(1 -y )x^(2) - 4 ( 1 - y)^(2) ge 0`
Now, if x is real, then
`D ge 0`
`rArr ( 1 + y)^(2) - 4 (1 - y )^(2) ge 0`
or `(1 + y - 2 + 2y) ( 1 + y + 2 - 2y) ge 0`
or `(3y - 1) ( 3 - y) ge 0`
or ` 3(y - (1)/(2))^(2 )(y - 3) le 0`
or `(1)/(3) le y le 3`
Hence, the range is `[1//3, 3)].`