1.

Find all possible values of the following expressions : `(i) sqrt(x^(2)-4) (ii) sqrt(9-x^(2)) (iii) sqrt(x^(2)-2x+10)`

Answer» (i) `sqrt(x^(2)-4)`
Least value of square root is 0 when `x^(2)=4 " or " x= +-2.` Also `x^(2)-4 ge 0`
Hence, `sqrt(x^(2)-4) in [0, oo).`
(ii) `sqrt(9-x^(2))`
Least value of square root is 0, when `9-x^(2)=0` or `x= +-3.`
Also, the greatest value of `9-x^(2)` is 9, when `x = 0`.
Hence, we have `0 le 9 - x^(2) le 9 impliessqrt(9-x^(2)) in [0,3].`
(iii)`sqrt(x^(2)-2x+10)=sqrt((x-1)^(2)+9)`
Here, the least value of `sqrt((x-1)^(2)+9)` is 3 , when `x-1=0`.
Also `(x-1)^(2)+9 ge 9impliessqrt((x-1)^(2)+9) ge 3`
Hence, `sqrt(x^(2)-2x+10) in [3, oo).`


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