Originally Posted by crounsa810
Combining functions is when you do g o f (x). g o f (x) = g (f(x)). So you gotta calculate f(x) first and then that will be your x in your g(x). You gotta be careful when you combine functions. The range of the first function must be in the domain of the second. For example if
f(x) = -x² - 5 (always negative)
g = square root function
g o f doesn't exist in that situation, because g (something negative) is impossible. So whenever you have a g o f (x) thing, always make sure it's legit.
Inverse Functions and their Representations
f(x) = y
The inverse function is the function (let's call it g) that will do g(y) = x. It doesn't always exist. For every y, there must be one and only one x.
For example if f(x) = x², f doesn't have an inverse function
f(3) = 9
f(-3) = 9
So for 9, you have 2 possible numbers. So it's not invertible.
In terms of composition, you have g o f = f o g = Id. Id is the function h(x) = x.
Again you have to be careful about the range and the domain of those functions. The domain of one must be the range of the other one.
Exponential Functions and Models
Exponential is the function that's the answer to the differential equation y' = y. So when you derive that function, you have the exact same thing.
exp(x) = 1 + x + x²/2 + (x^3)/6 + ... + (x^n) / (n!) + ... (goes on forever)
Derive that, you get the exact same thing.
It's also e^x, where e is a real number, so it has all the properties you could expect from it.
exp(a+b) = exp(a) * exp(b).
It's no different than when you do x² * x^3 = x^5
x² * x^3 = x^(2+3) = x^5
exp(a-b) = exp(a)/exp(b)
Because a-b = a + (-b). And x^(-n) = 1/(x^n)
Logarithmic Functions and Models
Log is the inverse function of exp.
ln(e^x) = x
And when you derive it, you get the function x --> 1/x
And one piece of advice, f and f(x) are not the same thing. f(x) is a number, f is a function. So never say g(x) is the inverse function of f(x), that doesn't make any sense.