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You can put this solution on your website I'll give you a hint to get you started If this doesn't help, either repost or email me (f+g) (x) is shorthand notation for f (x)+g (x) So (f+g) (x) means that. (f+g) (x) = f (x)+g (x) is the definition of the function (f+g) With this definition, polynomials form a vector space. What is the property where f (g (x)) = g (f (x))?besides being called (composition) commutative, it is sometimes also said that such functions are permutable, e.g You'll need to complete a few actions and gain 15 reputation points before being able to upvote Upvoting indicates when questions and answers are useful Regarding the question, if h (x) is a composite function such that h (x)=f (g (x)), then what does derivative of f (g (x)) w.r.t g (x) mean? the most common context is the chain rule $\cdot$ denotes multiplication, but i cannot be sure you mean $ (f\times g) (x)$. F (x) = x^3 and g (x) = x^2 both compositions (going f (g (x)) and g (f (x)) yield even results) however, when i use the trig functions, something different happens F (x) = sin (x) g (x) = cos (x). However, at values that don't occur in the range of g g (i.e For values y y which don't occur as y = g(x) y = g (x) for any x x), the function f f could take any arbitrary value, and we would not be. This originated when a math teacher said that in order for $g$ of $x$ to be the inverse of $f$ of $x$, it must be that both $f\circ g$ and $g \circ f$ equal $x$, and a student.