Question 95480
<pre><font size = 4><b>
Please help me find the inverse of the function.
{{{f(x)= (-5+5x)/(3-x)}}}

The graph of the inverse of a function is the reflection
of the graph across the line whose equation is y = x which
is called the "identity line", because y and x are 
identically equal on that line.

So we'll be able to check that when we finish, we first
graph the equation by getting points, plotting them and
drawing a smooth curve through them. 

{{{f(x)= (-5+5x)/(3-x)}}}

{{{drawing(600,562.5,-20,20,-20,20,

 graph(700,656.25,-20,20,-20,20,(-5+5x)/(3-x)*sqrt(3-x)/sqrt(3-x)),
graph(700,656.25,-20,20,-20,20,(-5+5x)/(3-x)*sqrt(x-3)/sqrt(x-3))

 )}}}

Now we follow the rules for finding the inverse:

1. Replace f(x) by y
2. Interchange x and y
3. Solve for y
4. Replace y by f<sup>-1</sup>(x)

1. Replace f(x) by y

{{{f(x)= (-5+5x)/(3-x)}}}
{{{y = (-5+5x)/(3-x)}}}

2. Interchange x and y

{{{x = (-5+5y)/(3-y)}}}

3. Solve for y

{{{x = (-5+5y)/(3-y)}}} 

Multiply both sides by {{{(3-y)}}}

{{{x(3-y) = -5+5y}}}

{{{3x-xy = -5+5y}}}

Get all terms in y on the right and
all others on left:

{{{3x+5=5y+xy}}}

Factor out y on the right

{{{3x+5=y(5+x)}}}

Divide both sides by {{{(5+x)}}}

{{{(3x+5)/(5+x) = y}}}

Swap sides for convenience 

{{{y =(3x+5)/(5+x)}}}

4. Replace y by f<sup>-1</sup>(x)

f<sup>-1</sup>(x) = {{{(3x+5)/(5+x)}}}

Now let's graph this on the same set of
axes in green:

{{{drawing(600,562.5,-20,20,-20,20,

 graph(700,656.25,-20,20,-20,20,(-5+5x)/(3-x)*sqrt(3-x)/sqrt(3-x)),
graph(700,656.25,-20,20,-20,20,(-5+5x)/(3-x)*sqrt(x-3)/sqrt(x-3)),

graph(700,656.25,-20,20,-20,20,0,(5+3x)/(5+x)*sqrt(x+5)/sqrt(x+5)),

graph(700,656.25,-20,20,-20,20,0,(5+3x)/(5+x)*sqrt(-x-5)/sqrt(-5-x))

 )}}}

Now we'll draw in the identity line whose equation is y = x. 
We'll draw it dotted just to see if the green graph is the 
reflection of the dark red one in that dotted identity line:

{{{drawing(600,562.5,-20,20,-20,20,

 graph(700,656.25,-20,20,-20,20,(-5+5x)/(3-x)*sqrt(3-x)/sqrt(3-x)),
graph(700,656.25,-20,20,-20,20,(-5+5x)/(3-x)*sqrt(x-3)/sqrt(x-3)),

graph(700,656.25,-20,20,-20,20,0,(5+3x)/(5+x)*sqrt(x+5)/sqrt(x+5)),

graph(700,656.25,-20,20,-20,20,0,(5+3x)/(5+x)*sqrt(-x-5)/sqrt(-5-x),x*sqrt(sin(4x))/sqrt(sin(4x))  )

 )}}}

Yes it is, so f<sup>-1</sup>(x) = {{{(3x+5)/(5+x)}}} must be right. 

Edwin</pre>