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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
4 years, 1 month 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 $$f(1) - f(-1) = -2, 0 ,2$$

- 4 years, 1 month ago

Such elegant solution!

- 4 years, 1 month ago