TMUA Logic Drill · Necessary and Sufficient

Question 7 · Hard · Paper 2 Style

The function f(x) is defined for all real numbers.

Consider the following three conditions, where a is a real constant:

I. f(a − x) = f(a + x) for all real x.

II. f(2a − x) = f(x) for all real x.

III. f(a − x) = f(x) for all real x.

Which of these conditions is/are necessary and sufficient for the graph of y = f(x) to have reflection symmetry in the line x = a?

A. I only B. II only C. III only D. I and II only E. I and III only F. II and III only G. I, II and III H. None of them

Correct answer: D

Solution:

For reflection symmetry in the vertical line x = a, points the same horizontal distance to the left and right of a must have equal function values.

Condition I: f(a − x) = f(a + x). These two inputs are equally spaced around a, so this is both necessary and sufficient.

Condition II: f(2a − x) = f(x). The points x and 2a − x are reflections of each other in the line x = a, so this is also both necessary and sufficient.

Condition III: f(a − x) = f(x). These points are generally symmetric about x = a/2, not about x = a. So III is not necessary and sufficient for symmetry in x = a.