Answer to Question #252425 in Algebra for gael

Question

If 𝑓(𝑥)=1𝑥, show that 𝑓(𝑎)−𝑓(𝑏)=𝑓(𝑎𝑏/𝑏−𝑎).

Answer 1

Solution:

Given $f(x)=\dfrac1x$

$LHS=f(a)-f(b) \\=\dfrac1a-\dfrac1b \\=\dfrac{b-a}{ab}$

$RHS=f(\dfrac{ab}{b-a}) \\=\dfrac{1}{\dfrac{ab}{b-a}} \\=\dfrac{b-a}{ab}$

So, LHS=RHS

Hence, proved.