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Let f(x) = [x]² + [x + 1] - 3, where [x] denotes the greatest integer less than or equal to x. Then which of the following is true?

1. f(x) is a many-one and into function
2. f(x) = 0 for infinitely many values of x
3. f(x) = 0 for exactly two real values
4. Both (a) and (b)

Correct answer: Both (a) and (b)

Solution

Let n=[x]. Then f = [x]^2 + [x+1] - 3 = n^2 + (n+1) - 3 = n^2 + n - 2 = (n+2)(n-1). This is 0 when n=-2 or n=1, i.e. for all x in [-2,-1) or [1,2): infinitely many values. The function is also many-one and into. Both (a) and (b) hold.

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