example of a composite function. Explain your variables how they're represented in the function, and which elem. functions are combined to form the composite function

Expert Answers
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You need to know that the process of composition of two functions is represented by the following notation(left) and the following result(right), such that:

`(fog)(x) = f(g(x))`

You should remember that if a problem requests to find the value of a function f(x), at a point x = a, you need to substitute a for x in equation of the function.

The same process works in composition of two function, thus, if the problem requests to find f(g(x)), you need to substitute g(x) for x in equation of f(x).

Since an worked example will make you better understand the process of composition, you may consider the following two functions, `f(x) = x^2`  and `g(x) = 7x+5` .

You need to find `(fog)(x)`  such that:

`(fog)(x) = f(g(x)) ` (first you need to write the result of composition)

`f(g(x))= (g(x))^2`  (substitute g(x) for x in equation of f(x))

`f(g(x)) = (7x+5)^2`  (replace g(x) with its equation)

`f(g(x)) = 49x^2 + 70x + 25`  (raise the binomial to square using the formula `(a+b)^2 = a^2 + 2ab + b^2` )

Hence, evaluating the result of composition of the functions f(x) and g(x) yields `f(g(x)) = 49x^2 + 70x + 25` .

Thus, the compositions of functions use the rules of substitution, as it is shown in the previous example, hence `(fog)(x) = f(g(x)).`