Question 348441
{{{f(x)=(x+3)/(x-2)}}}
<pre>
The domain if f(x) is 

{{{"("}}}{{{infinity}}}{{{","}}}{{{"2)"}}}{{{U}}}{{{"(2,"}}}{{{infinity}}}{{{")"}}}


{{{drawing(400,400,-10,10,-10,10,

graph(400,400,-10,10,-10,10, (x+3)/(x-2)) )}}}


Replace f(x) for y

{{{y=(x+3)/(x-2)}}}

Interchange x and y:

{{{x=(y+3)/(y-2)}}}

Solve for y

{{{x(y-2)=y+3}}}

{{{xy-2x=y+3}}}

{{{xy-y=2x+3}}}

{{{y(x-1)=2x+3}}}

{{{y=(2x+3)/(x-1)}}}

chnage y to f<sup>-1</sup>(x)

{{{f^(-1)}}}{{{(x)=(2x+3)/(x-1)}}}

The domain is {{{"("}}}{{{infinity}}}{{{","}}}{{{"1)"}}}{{{U}}}{{{"(1,"}}}{{{infinity}}}{{{")"}}} 

The green graph is the inverse.

{{{drawing(400,400,-10,10,-10,10,

graph(400,400,-10,10,-10,10, (x+3)/(x-2),(2x+3)/(x-1)) )}}}

Notice that the inverse f<sup>-1</sup>(x) is the reflection of f(x) 
across the identity line, the line whose equation is y=x, a line through
the origin that bisects to 1st and 3rd quadrants, the dotted line below:

{{{drawing(400,400,-10,10,-10,10,

graph(400,400,-10,10,-10,10, (x+3)/(x-2),(2x+3)/(x-1),

x*sqrt(sin(10x))/sqrt(sin(10x))


) )}}}

Edwin</pre>