1.

Let `f:RtoR:f(x)=x^(2)+1`. Find (i) `f^(-1){-4}` (ii) `f^(-1){10}` (iii) `f^(-1){5,17}`.

Answer» It is given that `f(x)=x^(2)+1`.
(i) Let `f^(1)(-4)=x`. Then,
`f(x)=-4impliesx^(2)+1=-4impliesx^(2)=-5`.
But, there is no real value of x whose square is -5.
`:.f^(1){-4}=phi`.
(ii) Let `:.f^(1)(10)=x`.
`f(x)=10impliesx^(2)+1=-4impliesx^(2)=9impliesx=pm3`.
`:.f^(1){10}={-3,3}`.
(iii) Let `f^(1)(5)=x`. Then,
`f(x)=5impliesx^(2)+1=5impliesx^(2)=4impliesx=pm2`.
`:.f^(1){5}={-2,2}`.
Let `f^(1)(17)=x`. Then,
`f(x)=17impliesx^(2)+1=17impliesx^(2)=16impliesx=pm4`.
`:.f^(1){17}={-4,4}`.
Hence, `f^(1){5,17}=-{-2,2,-4,4}`.


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