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TMUA 2018 · Paper 2 · Question 10 of 20

TMUA 2018 Paper 2 Question 10

Logic and arguments — Necessary and sufficient conditions for reflection symmetry. Try it first; the answer and a full worked solution are below.

TMUA 2018 · Paper 2Logic and argumentsNecessary and sufficient conditions for reflection symmetry8 options
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?

Each option below gives an answer of yes or no for each of the three conditions, in the order I, II, III.

  1. AI yes, II yes, III yes
  2. BI yes, II yes, III no
  3. CI yes, II no, III yes
  4. DI yes, II no, III no
  5. EI no, II yes, III yes
  6. FI no, II yes, III no
  7. GI no, II no, III yes
  8. HI no, II no, III no
Show the answer and worked solution
answer · B
  1. AI yes, II yes, III yes
  2. BI yes, II yes, III no
  3. CI yes, II no, III yes
  4. DI yes, II no, III no
  5. EI no, II yes, III yes
  6. FI no, II yes, III no
  7. GI no, II no, III yes
  8. HI no, II no, III no
Reflection in x = a sends the point at a + x to the point at a − x, so the symmetry says exactly f(a+x) = f(a−x): condition I is a restatement of it, hence both necessary and sufficient. Condition II is the same thing in different letters — replace x by a + t and it becomes f(a − t) = f(a+t) — so it works too. Condition III pairs x with a − x, whose midpoint is a2, so it describes symmetry in x = a2 instead; take a = 2 and f(x) = x2 to see the two come apart.