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f(x)={(x", if x is rational"),(0", ...

`f(x)={(x", if x is rational"),(0", if x is irrational"):}, g(x)={(0", if x is rational"),(x", if x is irrational"):}` Then, `(f-g)` is (A) one-one and onto (B) one-one but nor onto (C) onto but not one-one (D) neither one-one nor onto

A

one-one and into

B

one-one but nor onto

C

onto but not one-one

D

one-one nor onto

Text Solution

Verified by Experts

The correct Answer is:
D
Let `phi (x)=f(x)-g(x)={(x", "x in Q),(-x", "x notin Q):}`
Now, to check one-one.
Take any straight line parallel to X-axis which will intersect `phi(x)` only at one point.
`rArr phi(x)` is one-one.
To check onto.
As `f(x)={(x", "x in Q),(-x", "x notin Q):}`, which shows
y = x and y = -x for rational and irrational values `rarr y in ` real numbers.
`therefore="Codomain"=rArr "onto" `
Thus, `f-g` is one-one and onto.
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