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In addition to the revision notes for Composite Functions on this page, you can also access the following Functions learning resources for Composite Functions
Tutorial ID | Title | Tutorial | Video Tutorial | Revision Notes | Revision Questions |
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16.4 | Composite Functions |
In these revision notes for Composite Functions, we cover the following key points:
By definition, a composite function is a new function obtained when one function is used as the input value for another function. Essentially, the output of the inner function (the function used as the input value) becomes the input of the outer function (the resulting value).
If f(x) acts the input of another function g(x), then we express the resulting composite function as g[f(x)] or g ◦ f(x). We read it as "the function g of f(x)" or "the function g around f(x)".
Likewise, if g(x) acts the input of another function f(x), then we express the resulting composite function as f[g(x)] or f ◦ g(x). We read it as "the function f of g(x)" or "the function f around g(x)".
We consider as composite functions not only those obtained when two different functions f(x) and g(x) are combined with each other, but also when a function f(x) is combined with itself, i.e. f ◦ f(x); when a function f(x) is multiplied or divided by a constant, i.e. c · f(x); when a function f(x) is raised to a certain power n, i.e. [f(x)]n, etc.
Composite functions have the following properties:
Evaluating a composite function means finding the value of that composite function for a given input x.
It is possible to compose a function with itself. Suppose f(x) is a function, then the composition of function f with itself will be (f ∘ f) (x) = f [f(x)].
We can also find the value of a composite function by considering the two original functions one by one. Thus, if we want to find f ◦ g (x) for x = a, then we find first g(a) = b, then we calculate f(b) (a and b are numbers).
The composition of functions f(x) and g(x) is schematically shown below.
We can find the value of a composite function at a given point even when the individual functions f(x) and g(x) are not explicitly given but only shown on the graph. In this case, we can use the information provided in the graph as a shortcut for finding values of each function or the resulting composite functions without having the need to know the exact formulas of the given functions. When doing this, we must take into account the fact that if the ordered pair (x, y) is a point of the function f(x), then f(x) = y. Given this, we follow the procedure below to calculate f ◦ g(a) for x = a:
Sometimes the information about functions is given by tables. So we apply the same procedure as when dealing with graphs. The only difference is that now we have to identify the information from the table, not from the graph.
If we have two functions g : X → Y and f : Y → Z then f ∘ g : X → Z. This means the domain of f ∘ g is X and its range is Z.
The procedure described below is used to determine the domain of a composite function f ◦ g when the two individual functions are defined algebraically.
The range R of a composite function is identified in the same way as for simple functions, i.e. by checking for any restrictions that give not allowed values. However, in composite functions this procedure is carried out after making all operations, that is after obtaining the final form of the function.
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