f(x) = (x+2)/(x+6)
What is f^(-1)(-6)? (what is f inverse of -6)
I found f' which was [(x+6)-(x+2)]/(x+6)^2
I set the function equal to -6 and found x to be -30/7.
Used the formula to find inverse function: 1/f'(f^(-1)(-6))
So I replaced f^(-1)(-6) with -30/7 and then the new equation is 1/f'(-30/7)
I got 0.0816327 which is incorrect. Help?
f^-1 of (-6) is the value of x that makes f(x) = -6
(x+2)/(x+6) = -6
x +2 = -6x - 36
7x = -38
x = -5.428
Check: x+2/(x+6) = -3.428/0.572 =-6
I think I did wrong math and ended up doing more than I was supposed to. Thanks for the help, I get it now!
To find the inverse of a function, we follow these steps:
1. Begin with the given function, f(x).
2. Replace f(x) with y.
3. Swap the variables x and y.
4. Solve the resulting equation for y.
5. Replace y with f^(-1)(x) to represent the inverse function.
Let's apply these steps to your function f(x) = (x+2)/(x+6):
1. Start with f(x) = (x+2)/(x+6).
2. Replace f(x) with y: y = (x+2)/(x+6).
3. Swap the variables x and y: x = (y+2)/(y+6).
4. Solve for y:
- Multiply both sides by (y+6):
x(y+6) = (y+2).
- Distribute on the left side:
xy + 6x = y + 2.
- Move all terms involving y to one side:
xy - y = 2 - 6x.
- Factor out y on the left side:
y(x - 1) = 2 - 6x.
- Divide both sides by (x - 1):
y = (2 - 6x)/(x - 1).
5. Replace y with f^(-1)(x) to represent the inverse function:
f^(-1)(x) = (2 - 6x)/(x - 1).
Now, let's find f^(-1)(-6) by substituting x = -6 into the inverse function:
f^(-1)(-6) = (2 - 6(-6))/(-6 - 1)
= (-34)/(-7)
= 34/7
Therefore, f^(-1)(-6) is equal to 34/7 or approximately 4.857142857.