Solveeit Logo
Login

Question

Question: If for non-zero x, \(af(x)+bf(\dfrac{1}{x})=\dfrac{1}{x}-5\) where a not equal to b, then find f(x)....

If for non-zero x, af(x)+bf(1x)=1x−5af(x)+bf(\dfrac{1}{x})=\dfrac{1}{x}-5 where a not equal to b, then find f(x).

Explanation

Solution

Replace x with 1x\dfrac{1}{x} in the given equation and then equate to find the answer.

Complete step-by-step answer:
Given,
af(x)+bf(1x)=1x−5af(x)+bf(\dfrac{1}{x})=\dfrac{1}{x}-5………(1)
On replacing x by 1/x in the equation we get,
Therefore, after replacing we get,
⇒af(1x)+bf(x)=x−5\Rightarrow af(\dfrac{1}{x})+bf(x)=x-5………. (2)
Now for the next step we multiply eq (1) with a and eq (2) with b we get,
⇒a2f(x)+abf(1x)=ax−5a\Rightarrow a^2f(x)+abf(\dfrac{1}{x})=\dfrac{a}{x}-5a………… (3)
⇒abf(1x)+b2f(x)=bx−5b\Rightarrow abf(\dfrac{1}{x})+b^2f(x)= \dfrac{b}{x}-5b……. (4)
Now for the next part of the solution we.
Subtract (4) from (3), on doing that we get,
⇒(a2−b2)f(x)=ax−bx−5a+5b\Rightarrow (a^2-b^2)f(x)= \dfrac{a}{x}-\dfrac{b}{x}-5a+5b
Now on simplifying we get,
⇒(a−b)(a+b)f(x)=(a−b)1x−5(a−b)\Rightarrow (a-b)(a+b)f(x)=(a-b) \dfrac{1}{x}-5(a-b)
On taking f(x) as the subject of the formula we get,
⇒f(x)=(1−5xx(a+b))\Rightarrow f(x)=(\dfrac{1-5x}{x(a+b)})
Before this is the answer to the question,
Therefore, f(x)=(1−5xx(a+b))f(x)=(\dfrac{1-5x}{x(a+b)}) is the correct answer.

Note: Students often make mistakes in the every step of the question. They are not able to substitute x with the correct variable in the equation. Put correct value while moving to the next step.