Find f(-1) (inverse) if y=4x+2/3x-1 and x is not equal with 1/3

Before we find , let's think for a minute about what  does, and what it means to find the inverse of a function. A function  essentially takes a value  from the domain and maps it to another value  in that function's range according to a set of rules. Here, those rules are defined by the formula



All the inverse does is swap the domain and range of the function. Now, instead of trying to map  to , we're trying to find a set of rules that'll map  back to . To find those rules, all we have to do is solve the above equation for .

First, we'll multiply both sides of the equation by  to get it out of the denominator:



Cancelling on the right side and distributing on the left, we get:



Next, we collect all of our x terms on one side, and all of our non-x terms on the other:



Let's rearrange the right side and factor out an x on the left:



And finally, we divide both sides by  to obtain our



This equation gives us the "rules" for mapping any given y in the range back to an x in the domain. If we swap the domain and the range, we can define our function