Solution:
Consider the following injective function:
[tex]f(x)=\frac{3x+6}{x-5}[/tex]
this is equivalent to:
[tex]y=\frac{3x+6}{x-5}[/tex]
by cross-multiplication, this is equivalent to:
[tex](x-5)y\text{ = 3x+6}[/tex]
now, applying the distributive law, we get:
[tex]xy-5y\text{ = 3x+6}[/tex]
this is equivalent to:
[tex]xy\text{ -5y -3x = 6}[/tex]
Applying the common factor, we get:
[tex]x(y-3)\text{ -5y = 6}[/tex]
this is equivalent to:
[tex]x\text{ (y-3)}=\text{ 6+5y}[/tex]
solving for x, we get:
[tex]x\text{ = }\frac{6+5y}{y-3}[/tex]
so that, we can conclude that the correct answer is:
[tex]f^{-1}(x)=\frac{6+5x}{x-3}[/tex]
