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Find the rage of each of the following f...

Find the rage of each of the following functions. (i) `f(x)" "=" "2" "" "3x ," "x in R , x" ">" "0` (ii) `f(x)" "=x^2+" "2` , x is a real number. (iii) `f(x)" "=" "x` , x is a real number.

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(i) We have, `f(x)=2-3x,"where "x""inRandxgt0`.
Now, `xgt0implies3xgt0implies-3xlt0`
`implies-3x+2lt0+2implies2-3xlt2`
`impliesf(x)lt2impliesf(x)in(-oo,2)`.
Hence, range `(f)=(-oo,2)`.
(ii) We have, `g(x)=x^(2)+2,x""inR`.
Now, `x""inRimpliesx^(2)ge0impliesx^(2)+2ge0+2`
`impliesx^(2)+2ge2impliesg(x)ge2`
`impliesg(x)in[2,oo)`. Hence, range `(g)=[2,oo)`.
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