Functional Equation

**Winding Function** · December 24th 2008, 05:59 AM

Find all functions $f : \mathbb{R} - \left\{ -\frac{1}{3}, \frac{1}{3} \right\} \rightarrow \mathbb{R}$ that satisfy the identity

$f \left(\frac{x+1}{1-3x} \right) = x-f(x)$

**PaulRS** · December 24th 2008, 08:18 AM

Consider the change of variables: $ x : \to {\textstyle{{x - 1} \over {1 + 3x}}} $

Then we get: $ f\left( x \right) = {\textstyle{{x - 1} \over {1 + 3x}}} - f\left( {{\textstyle{{x - 1} \over {1 + 3x}}}} \right) $ ( check that the denominator is never 0)

So we've got these 2 equations: $ \left[ \begin{array}{l} f\left( x \right) = {\textstyle{{x - 1} \over {1 + 3x}}} - f\left( {{\textstyle{{x - 1} \over {1 + 3x}}}} \right) \\ f\left( {{\textstyle{{x + 1} \over {1 - 3x}}}} \right) = x - f\left( x \right) \\ \end{array} \right. $

Thus: $ f\left( {{\textstyle{{x + 1} \over {1 - 3x}}}} \right) - f\left( {{\textstyle{{x - 1} \over {1 + 3x}}}} \right) = x - {\textstyle{{x - 1} \over {1 + 3x}}} = {\textstyle{{3x^2 + 1} \over {3x + 1}}} $

Again take: $ x: \to {\textstyle{{x - 1} \over {1 + 3x}}} $

Then: $ f\left( x \right) - f\left( {{\textstyle{{x + 1} \over {1 - 3x}}}} \right) = {\textstyle{{3 \cdot \left( {{\textstyle{{x - 1} \over {1 + 3x}}}} \right)^2 + 1} \over {3 \cdot \left( {{\textstyle{{x - 1} \over {1 + 3x}}}} \right) + 1}}} $

Consider then the system of equations: $ \left\{ \begin{array}{l} f\left( x \right) - f\left( {{\textstyle{{x + 1} \over {1 - 3x}}}} \right) = {\textstyle{{3 \cdot \left( {{\textstyle{{x - 1} \over {1 + 3x}}}} \right)^2 + 1} \over {3 \cdot \left( {{\textstyle{{x - 1} \over {1 + 3x}}}} \right) + 1}}} \\ f\left( {{\textstyle{{x + 1} \over {1 - 3x}}}} \right) = x - f\left( x \right) \\ \end{array} \right. $

and solve for $ f\left( x \right) $

**Winding Function** · December 24th 2008, 09:11 AM

Originally Posted by PaulRS

and solve for $ f\left( x \right) $

How would I do this?

**PaulRS** · December 24th 2008, 09:12 AM

Originally Posted by Winding Function

How would I do this?

As you'd solve in a normal system of equations.

**Winding Function** · December 24th 2008, 04:41 PM

Originally Posted by PaulRS

As you'd solve in a normal system of equations.

I'm confused. - What are the "variables" of the equation, and how do I isolate them?

**Krizalid** · December 24th 2008, 05:23 PM

Originally Posted by Winding Function

I'm confused. - What are the "variables" of the equation, and how do I isolate them?

Originally Posted by PaulRS

and solve for $ f\left( x \right) $

**Winding Function** · December 24th 2008, 06:56 PM

Originally Posted by Krizalid

But what do I do with terms like $f\left( \frac{x+1}{1-3x} \right)$ ?

**Mathstud28** · December 24th 2008, 09:31 PM

Originally Posted by Winding Function

But what do I do with terms like $f\left( \frac{x+1}{1-3x} \right)$ ?

What happens if you add them?

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