I believe that the problem should read as follows:
Aman verma wrote:
Q: If the graph of the function y = f(x) is symmetrical about the line x = 2, then which of the following MUST be true?
(i) f(x + 2) = f(x - 2)
(ii) f(2 + x) = f(2 - x)
(iii)f(x) = f(-x)
(A) (i) only
(B) (ii)only
(C) (iii)only
(D) (i),(iii) Both
(E) (i),(ii) and (iii)
Symmetry about x=2 implies the following:
If the value of x is
k places TO THE LEFT OR RIGHT of x=2, the value of f(x) must be THE SAME in each case.
For example:
f(1) = f(3).
Here, the x-values (1 and 3) are each ONE PLACE away from x=2, so the value of f(x) must be the same in each case.
f(0) = f(4).
Here, the x-values (0 and 4) are each TWO PLACES away from x=2, so the value of f(x) must be the same in each case.
f(-1) = f(5).
Here, the x-values (-1 and 5) are each THREE PLACES away from x=2, so the value of f(x) must be the same in each case.
Implication:
x-values that are the SAME DISTANCE from x=2 MUST yield the same value for f(x).
x-values that are DIFFERENT DISTANCES from x=2 do NOT have to yield the same value for f(x).
(i) f(x + 2) = f(x - 2)
If x=1, we get:
f(1+2) = f(1-2).
f(3) = f(-1).
Doesn't work:
3 and -1 are DIFFERENT distances from x=2, so f(3) does not have to be equal to f(-1).
The correct answer choice cannot include (i).
Eliminate A, D and E.
(ii) f(2 + x) = f(2 - x)
If x=1, we get:
f(2+1) = f(2-1).
f(3) = f(1).
This works.
3 and 1 are each the SAME distance from x=2, so f(3) must be equal to f(1).
If x=10, we get:
f(2+10) = f(2-10).
f(12) = f(-8).
This works.
12 and -8 are each the SAME distance from x=2, so f(12) must be equal to f(-8).
The cases above illustrate that the correct answer choice must include (ii).
Eliminate C.
The correct answer is
B.
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