Let `f (x) =[0`, if x is rational and x if x is irrational and `g (x)=[0` , if x is irrational and x if x is rational then the function `(f-g)x` is

Options

A. one-one and into
B. Neither one-one nor onto
C. many-one and onto
D. one-one and onto

Answer

D

Explanation

Let `phi(x)=f(x)-g(x)={{:(" "x",",,x in Q),(-x",",,x!inQ):}`
For one-one
Take any straight line parallel to X-axis which will intersect `phi(x)` only at one point.
So, `phi(x)` is one-one.
For onto As, `phi(x)={{:(x",",,x inQ),(-x",",,x !inQ):}`,
which shows y = x and y = -x for rational and irrational values.
`implies" "y in "Real numbers."`
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So, `phi(x)` is onto.
Thus, f - g is one-one and onto.