Answer to Question #149509 in Algebra for Sourav Mondal

Question

Let A= R\ {1}, B= R\ {2} and f: A--> B

and g :B --> A be defined by f(x) = 2x /(x-1) and

g(x) = x/(x-2) .

Check whether f and g are functions. Compute go f. Are f and g invertible functions ? Justify your answer.

Answer 1

f(x) doesn't exist only if x-1=0, x=1 but point x=1 isn't a point of A, so yes, f is a function. g(x) doesn't exist only if x-2=0, x=2 but point x=2 is not in B, so is a function as well.

$g\circ f=g(f(x))=g(\frac{2x}{x-1})=\frac{2x}{x-1}:(\frac{2x}{x-1}-2)=$

$=\frac{2x}{x-1}:\frac{2x-2x+2}{x-1}=x$ ,

so if we act on element of A with f and then on its result we act with g we get the same element of A. This shows that not only g and f are invertible but they are each other's inversions - $f^{-1}=g, g^{-1}=f$. Both functions exist on their domains - A and B, so they qualify a definition of inversion