If g(X) = 2+ln(x-5), find the inverse function, (g^(-1))(x)
Solution:
y = 2+ ln(x-5)
e^y = 2 + x - 5
x = e^y + 3
y = e^x + 3
g^(-1) = y = e^x + 3
Is that right? I'm following an example - but don't really get it - how do I determine the domain and range - I know they flip when I do the inverse.
x = 2+ln(y-5)
x-2 = ln(y-5)
e^(x-2) = y-5
y = e^(x-2) + 5
Yes, you have correctly determined the inverse function, which is g^(-1)(x) = e^x + 3. To determine the domain and range of the inverse function, you can use the domain and range of the original function as a starting point.
For the original function g(x) = 2 + ln(x-5), the domain is restricted by the natural logarithm. Since the argument of the ln function must be greater than 0, we have x - 5 > 0, which implies x > 5. Therefore, the domain of g(x) is x > 5.
When finding the inverse function, the domain and range of the original function are swapped. So, for the inverse function g^(-1)(x) = e^x + 3, the domain will be x > 3 (based on the range of the original function) and the range will be all real numbers. The range of g^(-1)(x) is determined by the fact that e^x is always positive, so adding a positive constant does not change the fact that the range will still be all real numbers.
To summarize:
- Domain of g(x) is x > 5
- Range of g(x) is all real numbers
- Domain of g^(-1)(x) is x > 3
- Range of g^(-1)(x) is all real numbers