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Consider the function f: R → R defined by f(x) = {x|x| − 4, if x is rational; x|x| − √3, if x is irrational}. Then f(x) is:

1. one-one and onto
2. one-one and into
3. many-one and onto
4. many-one and into

Correct answer: one-one and into

Solution

The function f(x) is one-one because it assigns different outputs for different inputs, distinguishing between rational and irrational numbers. It is into because it does not cover all possible real numbers as outputs, specifically missing some values that could be produced by either case.

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