Question 118159
Change your life?
As long as there's no pressure. 
{{{f(x)=(-2)/(x+5)-1}}}
{{{g(x)=x-2}}}
To find the composite f(g(x)), substitute (x-2) where there is an x in f(x).
{{{f(g(x))=(-2)/((x-2)+5)-1}}}
{{{f(g(x))=(-2)/(x+3)-1}}}
The domain of f(g(x)) is all points except x=-3, since the denominator goes to zero.
For the range of f(x), as x gets large, either positive or negative, the term (-2)/(x+3) goes to zero and
{{{f(x)->-1}}}
Here is the graph of f(g(x)).
{{{ graph( 300, 300, -20, 20, -5, 5,(-2)/(x+3)-1) }}} 
That was f(g(x)).
Now for the inverse. 
To find the inverse of 
{{{y=(-2)/(x+3)-1}}}
Reverse x and y and solve for y.
{{{y=(-2)/(x+3)-1}}} Original equation of f(g(x)).
{{{highlight(x)=(-2)/(highlight(y)+3)-1}}} Interchange x and y.
{{{x(y+3)=-2-(y+3)}}} Multiply both sides by (y+3).
{{{xy+3x=-2-y-3}}} Distributive property.
{{{xy+3x=-y-5}}} Simplify.
{{{xy+y=-3x-5}}} Group like terms (y on left, x on right).
{{{y(x+1)=-(3x+5)}}}Distributive property.
{{{y=-(3x+5)/(x+1)}}}Divide by (x+1).
There it is. 
{{{f(g(x))^-1=-(3x+5)/(x+1)}}}
Domain of the inverse function is all numbers except x=-1. 
Think about the range of f(g(x)) and the domain of the inverse and you'll see why. 
Hint : You never reach -1 in f(g(x)) except at {{{infinity}}}
Range of the inverse function: as x gets large, the inverse function looks like,
{{{y=-(3x/x)=-(3cross(x)/cross(x))=-3}}} 
Again think about the range of the inverse and the domain of the f(g(x)) for reasons why. 
Here's the graph of the inverse. 
{{{ graph( 300, 300, -5, 5, -5, 5,-(3x+5)/(x+1))}}}