Question

Let f(x) be a real-valued function defined on: R → R such that f(x) = [x]² + [x + 1] −3, where [x] = the greatest integer ≤ x. Then 

(A)  f(x) is a many-one and into function

(B)  f(x) = 0 for infinite number of values of x 

(C)  f(x) = 0 for only two real values

(D)  none of these

Answer 1

Correct option  (A, B)

f(x) = [x]² + [x + 1] – 3 = {[x] + 2} {[x] – 1}

So, x = 1, 1⋅1, 1⋅2, ... ⇒ f(x) = 0

Therefore, f(x) is many one. Only integral values will be attained. Therefore, f(x) is into