Let f: R → R be a real-valued function such that 0 ≤ f(x) ≤ 1/2 for every x ∈ R. If, for a fixed a 0, the relation f(x + a) = 1/2 − √(f(x) − [f(x)]²) holds for all x ∈ R, then the period of f is

Question

Let f: R → R be a real-valued function such that 0 ≤ f(x) ≤ 1/2 for every x ∈ R. If, for a fixed a > 0, the relation f(x + a) = 1/2 − √(f(x) − [f(x)]²) holds for all x ∈ R, then the period of f is

Accepted Answer

2a. The relation given for f(x + a) indicates that the function's value at x + a is determined by its value at x, suggesting a repetitive behavior. By substituting x with x + a in the original equation, it can be shown that f(x + 2a) returns to f(x), confirming that the function is periodic with a period of 2a.

Let f: R → R be a real-valued function such that 0 ≤ f(x) ≤ 1/2 for every x ∈ R. If, for a fixed a > 0, the relation f(x + a) = 1/2 − √(f(x) − [f(x)]²) holds for all x ∈ R, then the period of f is

Solution

Related JEE Main Maths questions