If `alpha, beta(alpha,beta)` are the points of discontinuity of the function `f(f(x))`, where `f(x)=1/(1-x)`, then the set of values of a foe which the points `(alpha,beta)` and `(a,a^2)` lie on the same side of the line `x+2y-3=0` , is

Options

A. `(-3//2,1)`
B. `[-3//2,1]`
C. `[1,oo)`
D. `(-oo,-3//2]`

Answer

A

Explanation

We have
`f(x)=(1)/(1-x)`
Clearly, f(x) is defined for all `x in R -(1)}`
Now,
`f(f(x))=f((1)/(1-x))=(1)/(1-(1)/(1-x))=(x-1)/(x)`
We find that f(f(x)) is defined for all `x ne 0,1`
Now, `f(f(x))=f((x-1)/(x))=(1)/(1-(x-1)/(x))="x for all x"in 0,1`
Thus, the set of points of discontinuity is `{0,1}`
the line x+2y-3=0. Therefore,
`(0+2-3)(a+2a^(2)-3)gt0`
`Rightarrow 2a^(2)+a-3 lt 0`
`Rightarrow (2a+3) (a-1) lt 0 Rightarrow -3//2 lt a lt 1`
`Rightarrow a in (-3//2,1)`