1.

Let `f:RtoR`, defined by `f(x)={{:(1",",ifx""inQ),(-1",",ifx""inQ):}` Find (i) `f(1)/(2)` (ii) `f(0.34)` (iii) `f(sqrt2)` (iv) `f(pi)` (v) range(f) (vi) `f^(-1)(1)` (vii) `f^(-1){1}`

Answer» Since each one of `(1)/(2)` and 0.34 is rational, we have
(i) `f((1)/(2))=1and(ii)f(0.34)=1`.
Since each one of `sqrt2andphi` is irrational, we have (iii) `f(sqrt2)=-1and(iv)f(pi)=-1`.
(v) range `(f)={f(x):x""inR}`
`={f(x):x""inQ}uu{f(x):x""inR-Q}={1,-1}`. (vi) `f^(1){1}={x:f(x)=1}=Q`.
(vii) `f^(1){-1}={x:f(x)=-1}=(R-Q)`.


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