inverse functions

Question

Given: f(x) = 

Find the inverse function, .

 = 

Answer 1

To find an inverse function always start by switching all your variables.

f(x) = 3x+(6/x)+2

rewrite with x and y:

y = 3x+(6/x)+2

no replace every y with an x and every x with a y:

x=3y+(6/y)+2

Now solve for y (note this is a challenging one because we have y in the numerator and denominator postions!)

xy=3y²+6+2y

0=3y²+(x+2)y+6 <-- not very pretty but what you have here is a quadratic!

a=3

b=(x+2)

c=6

y=(-b+sqrt(b²-4ac))/(2a), (-b-sqrt(b²-4ac))/(2a)

plug in for a, b, and c

f^-1(x)= (-x-2+sqrt( (x+2)²-4(3)(6) )) / (2*3)

= (-x-2+sqrt( x²+4x+4-72 )) / 6

= (-x-2+sqrt(x²+4x-68))/6

I'll let you work the algebra out on the second equation for the other half of the inverse. Good luck!