Question

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

Answer 1

Before we find
`f^(-1)`, let's think for a minute about what
`f` does, and what it means to find the inverse of a function. A function
`f` essentially takes a value
`x` from the domain and maps it to another value
`y` in that function's range according to a set of rules. Here, those rules are defined by the formula


`y= (4x+2)/(3x-1) , x \\eq (1)/(3)`

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

First, we'll multiply both sides of the equation by
`3x-1` to get it out of the denominator:


`(3x-1)y=( (4x+2)/(3x-1))(3x-1)`

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


`3xy-y=4x+2`

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


`(3xy-y)+y=(4x+2)+y\\(3xy)-4x=(4x+2+y)-4x\\3xy-4x=2+y`

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


`x(3y-4)=y+2`

And finally, we divide both sides by
`3y-4` to obtain our answer:


`[x(3y-4)]/(3y-4)=(y+2)/(3y-4)\\\\x= (y+2)/(3y-4)`

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


`f^(-1)(x)= (x+2)/(3x-4)`