Find the value of `x^(2)` for the following values of x:
(a) `[-5,-1] " (b) "(3,6) " (c ) "(-2,3]`
(d) `(-3,oo) " (e ) "(-oo,4)`

Answer

(a) `[1,5] " (b) "(9,36)" (c ) " [0,9] " (d) " [0, oo) " (e ) "[0, oo)`

Explanation

(a) `-5 le x le -1`
`implies x^(2) in [1,5]`
(b) `3 lt x lt 6`
(c ) `-2 lt x le 3`
`implies -2 lt x le 3`
For `-2 lt x lt 0, x^(2) in (0,4) " (1)" `
and for `0 le x le 3, x^(2) in [0,9] " (2) `
From (1) and (2), `x^(2) in [0,9] `
Alternatively, ` x in (-2,3],` now least value of `x^(2)` is 0 which occurs when `x=0`
Greatest value of `x^(2)` is 9 for `x=3`
`implies x^(2) in [0,9]`
(d) `(-3,oo)`
Here least value of `x^(2)` is 0 for `x = 0`, and when x goes up to infinity, `x^(2)` also goes up to infinity
`implies x^(2) in [0,oo)`
(e) `(-oo,4)`
Here least value of `x^(2)` is 0 for `x = 0` and `x^(2) to oo`, when `x to -oo`
Hence, `x^(2) in [0,oo)`