Question

PLEASE HELP 46 POINTS Use composition of functions to determine whether or not f(x) and g(x) are inverses of each other. Show all work for full credit.
f(x) = (2/x-6)

g(x) = (6x+2/x)

Answer 1

Remark

The reason you are not getting many replies to this is an uncertainty about f(x). We do not know if you mean f(x) = 2/(x - 6) or f(x) = (2/x) - 6. We'll try it both ways.

First Way f(x) = 2/(x - 6)

f(x = y = 2/(x - 6)             Interchange x and y

x = 2 / (y - 6)                   Multiply both sides by y - 6

x(y - 6) = 2                      Divide by x

y - 6 = 2/x                       Add 6 to both sides.

y  = 2/x + 6                     Which is close to g(x) but it is not the same thing, even if you use x as a common denominator. That would give you

$\dfrac{6x + 2}{x}$

Second Way: f(x) = (2/x) - 6

f(x) = y = (2/x) - 6           Interchange x and y

x = (2/y) - 6                    Add 6 to both sides

x + 6 = 2/y                     Multiply  both sides by y

y(x + 6) = 2                    Divide by sides by (x + 6)

y = 2/(x + 6)  

Comment

These do not look anything alike so g(x) and f(x) are not inverses.