Question

f ( x ) = 1 x − 6 , g ( x ) = 6 x + 1 x Use composition to prove whether or not the functions are inverses of each other. Express the domain of the compositions using interval notation.

Answer 1

Two functions f(x) and g(x) are inverses if and only if:

f( g(x) ) = x

and

g( f(x) ) = x

In this case, we have:

f(x) = 1*x - 6

g(x) = 6*x + 1

Let's check if the functions are inverses.

f( g(x) ) = 1*g(x) - 6 = 1*(6x + 1) - 6 = 6x + 1 - 6 = 6x - 5

and

g( f(x) ) = 6*f(x) + 1 = 6*(1x - 6) + 1 = 6x - 36 + 1 = 6x - 35

So we can see that:

f( g(x) ) ≠ x

g( f(x) ) ≠ x

Thus, f(x) and g(x) are not inverses.

Particularly, the two compositions are:

f( g(x) ) = 6x - 5

g( f(x) ) = 6x - 35

Both of these are linear functions, thus the domain in both cases is the set of all real numbers, that can be written as:

domain = (-∞, ∞)