inverse functions?

Question

f(x)= x+3/x+9

f^-1(-6)= ????



show work please!!

Answer 1

I'm not sure if you mean f(x) = (x+3)/(x+9) or f(x) = x + 3/x + 9.

For f(x) = (x+3)/(x+9), we need to solve
(x+3) / (x+9) = -6
=> x + 3 = -6(x+9)
=> 7x = -57
=> x = -57/7.
So f^-1(-6) = -57/7.

For f(x) = x + 3/x + 9, we need to solve
x + 3/x + 9 = -6
=> x^2 + 3 + 9x = -6x
=> x^2 + 15x + 3 = 0
This has solutions
x = [-15 ± √(225-12)] / 2
= (-15 ± √213) / 2.
For the inverse to exist, the domain of f must exclude exactly one of these values; then f^-1(-6) will be the other value.

Answer 2

Imagine F(x) to be y.
y=(x+3)/(x+9)
Replace y's for x's and x's for y's. Like this.
x=(y+3)/(y+9)
(y+9)x=(y+3)
xy+9x-y=3
xy-y=3-9x
y(x-1)=[3(1-3x)] Divide by (x-1)
f'(x)=[3(1-3x)]/(x-1)
f'(-6)=[3(1+18)]/(-7)
f'(-6)=-(57/7)
I hope that's right. You can check it if you want.