hey all, need help with a problem
Question: Use composition to show that f(x) = (x + 3) / (x - 2) and g(x) = (2x - 3) / (x - 1) are inverse of each other.
Any help would be greatly appreciated, thanks
-Laconian
What means that f and g are inverse of each other?
Just that $\displaystyle \forall x \in Dom(g), f(g(x))=x$ and $\displaystyle \forall x \in Dom(f), g(f(x))=x$
For example, let $\displaystyle x \in \mathbb{R}-\{1\},$
$\displaystyle f(g(x))$
$\displaystyle =f(\frac{2x-3}{x-1})$
$\displaystyle =\frac{\frac{2x-3}{x-1}+3}{\frac{2x-3}{x-1}-2}$
$\displaystyle =\frac{2x-3+3(x-1)}{2x-3-2(x-1)}$
$\displaystyle =\frac{5x-6}{-1}$ !!! that's not what we wanted!
But if we change $\displaystyle f(x)=\frac{x+3}{x-1}$ by $\displaystyle f(x)=\frac{x-3}{x-1}$, as you can see, that would work. Then try with $\displaystyle gof$.