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Question:

If f(x) + 2f(1/x) = 3x, x ≠ 0 and S = {x ∈ R: f(x) = f(-x)}, then S:

is an empty set.

contains exactly one element.

contains exactly two elements.

contains more than two elements.

Solution:

f(x) + 2f(1/x) = 3x
Putting x as 1/x then f(1/x) + 2f(x) = 3(1/x)
Solving these two equations,
f(x) = x - 1/x
Now for f(x) = f(-x) ⇒ x - 1/x = -x + 1/x
⇒ 2x = 2/x ⇒ x² = 1 ⇒ x = ±1