'Y' is a function 'f' of 'x': Y = f(x) . 'Z' is a function 'g' of 'y': Z = g [ f(x) ] .
No, it is not. f(x) = 2x + 3 and g(x) = 3x2 are polynomials but f(x)/g(x) is not a polynomial.
composite of a function is fog(x)=f(g(x))
It depends on the context. The additive inverse of a number, X, is the number -X such that their sum is 0. The multiplicative inverse of a (non-zero) number, Y, is the number -Y such that their product is 1. The inverse of a function f, is the function g (over appropriate domains and ranges) such that if f(X) = Y then g(Y) = X. So, for example, if f(X) = 2X then g(X) = X/2 or if f(X) = exp(X) then g(X) = ln(X), and so on.
If x = g(y) ∫ f(x) dx = ∫ f(g(y))g'(y) dy This is called change of variables.
Function "f" depends on "x", and function "g" depends on function "f".
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In the function G(F(x)), F is a function that relies on G, creating a circular dependency where G's output influences F's behavior. Simultaneously, G itself is dependent on the input x, indicating that changes in x will affect G's output. This interdependence can lead to complex relationships and potentially recursive behavior, depending on how F and G are defined. Care must be taken to ensure that such dependencies do not lead to infinite loops or undefined outcomes.
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'Y' is a function 'f' of 'x': Y = f(x) . 'Z' is a function 'g' of 'y': Z = g [ f(x) ] .
Provided that the range of g(x) is the domain of f(x) then it is the composite function, called f of g of x.Note that f(g(x) ) is not the same as g(f(x).For example, if f(x) = x + 2 and g(x( = 3*x for real x, thenf(g(x)) = f(3*x) = 3*x + 2while g(f(x)) = g(x + 2) = 3*(x + 2) = 3*x + 6
∫ [f'(x)g(x) - g'(x)f(x)]/[f(x)g(x)] dx = ln(f(x)/g(x)) + C C is the constant of integration.
if f(x) = 4x, then the inverse function g(x) = x/4
∫ [f'(x)g(x) - g'(x)f(x)]/g(x)2 dx = f(x)/g(x) + C C is the constant of integration.
You would have been given a function for f(x) and another function for g(x). When you want to find f(g(x)), you put the function for g(x) wherever x occurs in f(x). Example: f(x)=3x+2 g(x)=x^2 f(g(x))=3(x^2)+2 I'm not sure what you mean by address domain and range. They depend on what functions you're given.
∫ f'(x)g(x) dx = f(x)g(x) - ∫ f(x)g/(x) dx This is known as integration by parts.