Find f(x) from Functional Equation

Find $f(x)$ from Functional Equation

Question: If for non-zero $x$, $$ af(x)+bf\left(\frac1x\right)=\frac1x-5, $$ where $$ a\ne b, $$ then find $f(x)$.

Solution

Given: $$ af(x)+bf\left(\frac1x\right)=\frac1x-5 \quad \cdots (1) $$

Replace $x$ by $\frac1x$:

$$ af\left(\frac1x\right)+bf(x)=x-5 \quad \cdots (2) $$

Multiply equation (1) by $a$:

$$ a^2f(x)+abf\left(\frac1x\right)=\frac{a}{x}-5a \quad \cdots (3) $$

Multiply equation (2) by $b$:

$$ abf\left(\frac1x\right)+b^2f(x)=bx-5b \quad \cdots (4) $$

Subtract (4) from (3):

$$ (a^2-b^2)f(x)=\frac{a}{x}-bx-5a+5b $$

$$ (a-b)(a+b)f(x)=\frac{a}{x}-bx-5(a-b) $$

Since $a\ne b$,

$$ f(x)=\frac{\frac{a}{x}-bx-5(a-b)}{(a-b)(a+b)} $$

Hence, $$ \boxed{ f(x)= \frac{\frac{a}{x}-bx-5(a-b)} {(a-b)(a+b)} } $$

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