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.

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 = (-∞, ∞)

the answer is (-4,-4)

hope this ! : d

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