Question

For each pair of function f and g below, find f(g(x)) and g(f(x)). Then determine whether f and g are inverses of each other. Simplify your answers as much as possible. Assume your expressions are defined for all x in the domain of the composition.

1. f(x) = 1/3x, x not equal to 0



g(x) = 1/3x, x not equal to 0 what does f(g(x)) =, what does g(f(x)) =, Is f and g inverses of each other?

Answer 1

Answer: f and g ARE INVERSES of each other

Explanation:

If f and g are inverses of each other, then their composition will equal x.

`\text{Given:}\quad f(x)=(1)/(3x)\qquad \qquad g(x)=(1)/(3x)`

`f(g(x))\\\\f\bigg((1)/(3x)\bigg)=(1)/(3((1)/(3x)))\quad =(1)/((1)/(x))\quad =x\\\\\\\\\\g(f(x))\\\\g\bigg((1)/(3x)\bigg)=(1)/(3((1)/(3x)))\quad =(1)/((1)/(x))\quad =x`

Since their compositions both equal "x", they are inverses of each other