Math problem

Question

Given the one to one function f(x)=x^3-6/x^3+2 ,find f^-1(x)

Answer 1

Finding an inverse equation with a one to one ratio means that you want to change the independent variable to dependent, and the dependent to independent. Basically you exchange variables and solve for your new dependent variable.

 

Restate the original function as y=x³-6/x³+2.

 

Now exchange the variables.  x=y³-6/y³+2

 

Solve for your new y value.

 

 

Answer 2

f^-1(x) is an inverse function

 

To determine an inverse of a function we do the following.

1) Replace the f(x) with a y

2) Replace all x's with y's and the y with an x

3) Solve for y

4) Replace the y in 3) with f^-1(x)

 

So f(x) = x³-6

              -----

             x³+2

Therefore  y = x³-6

                      -----
                     x³+2

 

Switching the x's and y's

x = y³-6

     ------

     y³+2

 

Let's work through this to solve for y

 

x(y³+2) = y³-6

xy³ + 2x = y³ -6

y³(x-1) = -6-2x

y3 = -2(x+3)

       ----------

        (x-1)

 

y = [-2(x+3)/(x-1)]^(1/3) 

 

f^-1(x) = [-2(x+3)/(x-1)]^(1/3)

where the power 1/3 is equivalent to a cube root

 

Hope this helps