1.

Find the range of the following functions given by `f(x)=(3)/(2-x^(2)) " "` (ii) `f(x)=1-|x-2|` (iii) `f(x)=|x-3| " "` (iv) `f(x)=1+3 cos 2x`

Answer» We have, `f(x)=(3)/(2-x^(2))`
Let y=f(x)
Then, `y=(3)/(2-x^(2))rArr 2-x^(2)=(3)/(4)`
`rArr x^(2)=2-(3)/(y)rArrx=sqrt((2y-3)/(y))`
x assums real values, if `2y-3ge0 and ygt0 rArr y ge (3)/(2)`
`:.` Range of `f=[(3)/(2),oo)`
(ii) We known that, `|x-2|ge or rArr-|x-2|le0`
`rArr 1-|x-2|le1 rArrf(x)le1`
`:.` Range of `f=(-oo,1]`
(iii) We know, that, `|x-3|leorrArrf(x)ge0`
`:.` Rang of `f=0,oo]`
(iv) We hnow that, `-1le cos2xle1rArr-3le3cos2xle3`
`rArr 1-3le1+3cos2xle1+3rArr-2le1+3cos2xle1+3`
`rArr -2lef(x)le4`
`:.` Range of `f=[-2,4]`


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