True Or nd?

Assume $f(x)=x$ . Then $f(x+2)=x+2$ . Since $x\ne x+2$ , we have a function for which this equality is not true, therefore it is not true in all cases.

Let's say for the sake of argument assume f(x) is differentiable for all real x, and let y = x + 2 so that we obtain:

f(x) = f(y) (i).

Differentiating (i) with respect to x and y respectively yields:

f' (x) = f' (y) = 0 => f' (x) = A (ii)

and integrating (ii) produces f(x) = Ax + B (iii), where A & B are real arbitrary constants. If we now substitute the linear equation in (iii) back into the original functional equation, we now get:

f(x) = f(x+2) => Ax + B = A(x+2) + B => 0 = 2A => A = 0

which leaves f(x) = B, a constant function, as the only feasible solution. Hence, the graph of f(x) is not always the same as f(x+2) for all real x.