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Originally Posted by crounsa810
Combining Functions

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.
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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.
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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(ab) = exp(a)/exp(b)
Because ab = a + (b). And x^(n) = 1/(x^n)
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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.