f(x) = (5e^x-6)/(17e^x+13) find f^-1(x). then: the domain of f^-1 (x) is in the open interval (a, b) find a and b
Given
y=f(x)=(15e^x-6)/(17e^x+13)
To find the inverse, follow the following steps:
1. invert x and y
x=(15e^y-6)/(17e^y+13)
2. solve for y in terms of x
(17x-15)e^y=-6-13x
e^y=(6-13x)/(17x-15)
take log on both sides:
y = f-1(x) = ln((-6-13x)/(17x-15))
3. check if f(f-1(x))=x
4. plot both f(x) and f-1(x) on the same graph. They should be mirror images of each other about the y=x line.
I leave it to you as an exercise to find the domain of f-1(x).
The domain is the intersection of the domain of the numerator and denominator.
In addition, for a log function, the argument to the log function must be non-negative.