Question 206027
{{{f(x) = 2/x-4}}}
To find the inverse function of a function, replace x by y and y (or f(x) ) by x and solve for y
 
{{{x = 2/y-4}}}      add 4 to each side
{{{x+4 = 2/y}}}     change the left side to a fraction and flip both sides
{{{1/(x+4) = y/2}}}     multiply both sides by 2, then switch sides so y (the inverse of f(x) ) is on the left
{{{f^-1(x) = y = 2 / (x+4)}}}
 
Here is a graph of f (red) and its inverse (green, the vertical line at -4 should be ignored, it is not part of either function). Also, graphed is the line y=x ... a function and its inverse are always mirror images of one another across this line, which you can see for the green and the red here.
{{{graph(300,300,-10,4,-10,4,(2/x) - 4, 2/(x+4), x ) }}}
 
 
The domain of f is all real numbers except 0 (because x = x-0 is in the denominator makes f(x) undefined).
The range of f is all real numbers except -4 ... for -4 to be the value of y, {{{2/x}}} would have to be 0 which means x would have to be infinitely large, but no real number is infinitely large.
The domain of {{{f^-1}}} is all real numbers except -4 (because x+4 = -4+4 is in the denomiator and makes {{{f^-1}}} undefined). 
The range of {{{f^-1}}} is all real numbers except 0 ... for 0 to be the value of y, {{{2/(x-4)}}} would have to be 0 which means x-4 (and therefore x) would have to be infinitely large, but no real number is infinitely large.