Question

A. Use composition to prove whether or not the functions are inverses of each other. B. Express the domain of the compositions using interval notation.

Answer 1

Given:
`f(x) = (1)/(x-2)`

`g(x) = (2x+1)/(x)`

A.)Consider

`f(g(x))= f((2x+1)/(x) )`

`f((2x+1)/(x) )=(1)/(((2x+1)/(x))-2)`

`f((2x+1)/(x) )=(1)/((2x+1-2x)/(x))`

`f((2x+1)/(x) )=(x)/(1)`

`f((2x+1)/(x) )=1`

Also,

`g(f(x))= g((1)/(x-2) )`

`g((1)/(x-2) )= (2((1)/(x-2)) +1 )/((1)/(x-2))`

`g((1)/(x-2) )= ((2+x-2)/(x-2) )/((1)/(x-2))`

`g((1)/(x-2) )= (x )/(1)`

`g((1)/(x-2) )= x`

Since,
`f(g(x))=g(f(x))=x`

Therefore, both functions are inverses of each other.

B.

For the Composition function
`f(g(x)) = f((2x+1)/(x) )=x`

Since, the function
`f(g(x))` is not defined for
`x=0`.

Therefore, the domain is
`(-\infty,0)\cup(0,\infty)`

For the Composition function
`g(f(x)) =g((1)/(x-2) )=x`

Since, the function
`g(f(x))` is not defined for
`x=2`.

Therefore, the domain is
`(-\infty,2)\cup(2,\infty)`