24.1.6. Composition of Functions

Now the last thing that we want to now define is the composition of functions. So, remember functions are special relations. So, since we can compose relations, we can compose functions as well provided certain conditions are satisfied. So, imagine you are given two functions, a function g and the function f with appropriate domain and co-domain. Then the composition of the function f and g is denoted by f∘g.

The composition of functions refers to combining two functions to create a new function. When you have two functions, g and f, you can apply them in sequence. The composition is written as f∘g (read as 'f composed with g'). This means that you first apply the function g and then apply the function f to the result from g. It's important that the output of g is compatible with the input requirements of f.

Think of a composition of functions like a recipe. For instance, consider a fruit smoothie recipe that requires you to first blend fruits (function g) and then chill the smoothie (function f). You can’t chill the smoothie until you’ve blended the fruits. Similarly, in mathematics, you need to complete the first function (g) before moving on to the second function (f).

So, let me make it more clear here. So, you imagine that g: A → B where A = B or A ≠ B. And f: B → C. Then the composition of f∘g will be a function from the set A → C. And what exactly will be that function? So, we will apply the function g first and obtain the images or possible elements a from the set A. And then we will apply the function f on those resultant elements and that will be the mapping of the element A as per the composition of f and g function, f∘g.

For the composition f∘g to be valid, the output of the first function (g) must fall within the input range of the second function (f). Specifically, if g maps elements from set A to set B, f must take elements from set B and map them to set C. This ensures that every output of g can be used as an input for f, resulting in a new function that maps elements from A directly to C.

Using our previous smoothie analogy, imagine you have a step where you first ‘blend fruits’ (function g) which gives you a mixture in a jar (set B). Next, you take this jar and ‘chill it’ (function f), giving you a cold smoothie in a glass (set C). If you didn't blend the fruits first, you couldn’t move on to the chilling step. Thus, just like a recipe, the transition from A to C through B is essential for correct composition.

It is easy to see that since it has to be a function then the composition of f and g is defined only if range(g) ⊆ domain(f). So, what is exactly the range(g) means? So, since g is a mapping from I am assuming here that g: A → B, then A is called as the domain and B is called as the co-domain. Then what is this range set? It is the set of all images of various elements from the A set. That means this range is a subset.

For the composition of functions to be valid, the range of g (which includes all possible outputs when g is applied) must be a subset of the domain of f (which includes all possible inputs for f). If there are outputs from g that do not match any allowable inputs for f, then you cannot compose the functions. Therefore, checking if range(g) is part of domain(f) is crucial before applying the composition.

Returning to our smoothie analogy, if the recipe for chilling (function f) requires a specific type of jar (inputs in the domain of f), but the blender produces an incompatible jar (outputs in the range of g), you cannot move to the chilling step. Ensuring that the outputs from the blending step match what is needed for the chilling step is like ensuring the range of g fits within the domain of f.

And again, the composition of functions need not be commutative in the sense f∘g ≠ g∘f. First of all the composition of g∘f need not be defined at all, if f∘g is defined because A, B, and C sets could be arbitrary.

Composition of functions is not commutative, meaning that f∘g does not necessarily equal g∘f. This is primarily because the output of the first function (g) must be suitable for the second function (f). If their inputs and outputs do not align, you may not even be able to perform both compositions, or the outcomes could be entirely different. This reaffirms the importance of the sequence in which functions are applied.

Consider a situation where you first apply paint (function g) and then draw over it with a marker (function f). If you draw first and then paint over, the results will be very different; you may lose the drawing entirely if the paint is too opaque! This illustrates how the order of operations matters and that switching them will yield different outcomes, highlighting the non-commutative nature of function composition.

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