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

Explanation:

A). If a function 'f' is inverse of another function 'g',

Then f[g(x)] = x = g[f(x)]

In this question the given functions are,

f(x) =
`(1)/(x-3)` and g(x) =
`(3x+1)/(x)`

Then, f[g(x)] =
`(1)/((3x+1)/(x)-3)`

=
`(x)/(3x+1-3x)`

= x

Similarly, g[f(x)] =
`(((3)/(x-3))+1)/((1)/(x-3) )`

=
`(3+x-3)/(1)`

= x

Therefore, Both the functions are inverse of each other.

B). Domain of the compositions of these functions will be a set of all real numbers, (-∞, ∞)