A Functional Equation #1

Algebra Level 3

Suppose that f ( x ) f(x) is a rational function such that x ≠ 0 x \ne 0 . Find f ( − 2 ) f(-2) for the following equation.

3 f ( 1 x ) + 2 f ( x ) x = x 2 3 f\left(\frac{1}{x}\right)+\frac{2 f(x)}{x}=x^{2}


The answer is 3.35.

Deva Craig by Deva Craig , 22, USA

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1 solution

Tom Engelsman
Apr 24, 2021

We can determine the required rational function f ( x ) f(x) via the following system of equations:

3 f ( 1 x ) + 2 f ( x ) x = x 2 \Large 3f(\frac{1}{x}) + \frac{2f(x)}{x} = x^2 (i)

3 f ( x ) + 2 x f ( 1 x ) = 1 x 2 \Large 3f(x) + 2xf(\frac{1}{x}) = \frac{1}{x^2} (ii)

Solving (i) for f ( 1 x ) f(\frac{1}{x}) in terms of f ( x ) f(x) and substituting it into (ii) yields the following function:

f ( x ) = 3 5 x 2 − 2 x 3 5 ⇒ f ( − 2 ) = 67 20 . \Large f(x) = \frac{3}{5x^2} - \frac{2x^3}{5} \Rightarrow \boxed{f(-2) = \frac{67}{20}}.

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