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

$y= \frac{4x+2}{3x-1} , x \neq \frac{1}{3}$

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:

$(3x-1)y=( \frac{4x+2}{3x-1})(3x-1)$

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:

$[x(3y-4)]/(3y-4)=(y+2)/(3y-4)\\\\x= \frac{y+2}{3y-4}$

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 

$f^{-1}(x)= \frac{x+2}{3x-4}$