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How can I deal with Functional Equations

When I came across this problem from PMO: f(a)+1/f(b) = f(1/a) + f(b) Where f is defined for all real numbers except zero. What are the possible values of f(1) - f(-1)?

Furthermore, how would I attack problems regarding functional equations, especially if the basic techniques may not work (eg. zeroing f(x))?

Note by John Ashley Capellan
3 years, 11 months ago

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Putting \( a = 1, b = -1 \)

\( f(1) + \frac{1}{f(-1)} = f(1) + f(-1) \)

\( \frac{1}{f(-1)} = f(-1) \)

\( f(-1)^2 = 1 \)

\( f(-1) = \pm 1 \)

Putting \( a = -1, b = 1 \)

\( f(-1) + \frac{1}{f(1)} = f(-1) + f(1) \)

\( \frac{1}{f(1)} = f(1) \)

\( f(1)^2 = 1 \)

\( f(1) = \pm 1 \)

Now, When \( f(1) = f(-1) = 1, \)

\( f(1) - f(-1) = 0 \)

When \( f(1) = 1 , f(-1) = -1 \)

\( f(1) - f(-1) = 2 \)

When \( f(1) = -1, f(-1) = 1 \)

\( f(1) - f(-1) = -2 \)

When \( f(1) = -1 , f(-1) = -1 \)

\( f(1) - f(-1) = 0 \)

Therefore possible values of \( f(1) - f(-1) = -2, 0 ,2 \) Siddhartha Srivastava · 3 years, 11 months ago

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@Siddhartha Srivastava Such elegant solution! John Ashley Capellan · 3 years, 11 months ago

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