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Függvényfüggvény, also known as a composite function, is a function that results from applying one function to the output of another function. In essence, it's a function within a function.

The notation for a composite function is typically written as f(g(x)) or (f ∘ g)(x). This means that first, the function g(x) is evaluated, and then the result of g(x) is used as the input for the function f. The order of composition is crucial; f(g(x)) is generally not the same as g(f(x)).

To illustrate, consider two functions: f(x) = x² and g(x) = x + 1.

If we want to find f(g(x)), we substitute g(x) into f(x). So, f(g(x)) = f(x + 1) = (x + 1)².

Conversely, if we want to find g(f(x)), we substitute f(x) into g(x). So, g(f(x)) = g(x²) = x² + 1.

Clearly, (x + 1)² ≠ x² + 1 in general.

Composite functions are a fundamental concept in mathematics and have wide-ranging applications in calculus, algebra, and other areas. They allow for the construction of more complex functions from simpler ones and are essential for understanding concepts like derivatives and integrals of composed functions (chain rule).