Suppose A is the father of B and B is the father of C. Who will be A for C? A is the grandfather of C. Here, we see that there is a relation between A and B, B and C and also between A and C. This relation between A and C denotes the indirect or the composite relation. In this section, we will get ourselves familiar with composite functions. Composite functions show the sets of relations between two functions. Let us start to learn the composition of functions and invertible function.
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Composite Functions
Suppose f is a function which maps A to B. And there is another function g which maps B to C. Can we map A to C? The mapping of elements of A to C is the basic concept of Composition of functions. When two functions combine in a way that the output of one function becomes the input of other, the function is a composite function.
In mathematics, the composition of a function is a step-wise application. For example, the function f: A→ B & g: B→ C can be composed to form a function which maps x in A to g(f(x)) in C. All sets are non-empty sets. A composite function is denoted by (g o f) (x) = g (f(x)). The notation g o f is read as “g of f”.
Consider the functions f: A→B and g: B→C. f = {1, 2, 3, 4, 5}→ {1, 4, 9, 16, 25} and g = {1, 4, 9, 16, 25} → {2, 8, 18, 32, 50}. A = {1, 2, 3, 4, 5}, B = {16, 4, 25, 1, 9}, C = {32, 18, 8, 50, 2}.Here, g o f = {(1, 2), (2, 8), (3, 18), (4, 32), (5, 50)}.
The composition of functions is associative in nature i.e., g o f = f o g. It is necessary that the functions are one-one and onto for a composition of functions.
Browse more Topics under Relations And Functions
- Relations
- Functions
- Types of Relations
- Types of Functions
- Representation of Functions
- Algebra of Real Functions
- Cartesian Product of Sets
- Binary Operations
Invertible Function
A function is invertible if on reversing the order of mapping we get the input as the new output. In other words, if a function, f whose domain is in set A and image in set B is invertible if f-1 has its domain in B and image in A.
f(x) = y ⇔ f-1 (y) = x.
Not all functions have an inverse. For a function to have an inverse, each element b∈B must not have more than one a ∈ A. The function must be an Injective function. Also, every element of B must be mapped with that of A. The function must be a Surjective function. It is necessary that the function is one-one and onto to be invertible, and vice-versa.
It is interesting to know the composition of a function and its inverse returns the element of the domain.
f-1 o f = f -1 (f(x)) = x
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Solved Examples for You
Question 1: If f: A → B, f(x) = y = x2 and g: B→C, g(y) = z = y + 2 find g o f.
Given A = {1, 2, 3, 4, 5}, B = {1, 4, 9, 16, 25}, C = {2, 6, 11, 18, 27}.
Answer : g o f(x) = g(f(x))
g(f(1)) = g(1) = 2, g(f(2)) = g(4) = 6, g(f(3)) = g(9) = 11, g(f(4)) = g(16) = 18, g(f(5)) = g(25) = 27.
Question 2: Write the inverse of the above g o f.
Answer : (g o f) -1 = f-1(g-1(z))
f-1(g-1(z)) = f-1(g-1(2)) = f-1(1) = 1, f-1(g-1(6)) = f-1(4) = 2, f-1(g-1(11)) = f-1(9) = 3, f-1(g-1(18)) = f-1(16) = 4 & f -1(g-1(27)) = f-1(25) = 5.
Question 3: What does the composite function mean?
Answer: Composite function refers to one whose values we find from two specified functions when we apply one function to an independent variable and then we apply the second function to the outcome. And, also whose domain comprises of those values of the independent variable for which the outcome produced by the first function that is lying in the domain of the second.
Question 4: Are composite functions associative?
Answer: The composition of functions is constantly associative. It is a property that it inherits from the composition of relations. In an austere sense, we can build the composition g ∘ f only if f’s codomain will equal g’s domain. In a broader sense, it is adequate that the former be a subset of the latter.
Question 5: Why are composite functions important?
Answer: Composite functions are so general that we typically don’t think to brand them as composite functions. Nevertheless, they happen any time a change in one quantity creates a change in another which, in result, creates a change in a third quantity.
Question 6: What functions are not invertible?
Answer: Functions are non-invertible for the reason that when taking the inverse, the graph becomes a parabola that opens to the right that is not a function. A sideways opening parabola comprises two outputs for every input that is not a function by definition. Further, you can also make the function invertible by limiting the domain.
The example for onto function doesn’t qualify as a function in the first place. Does it??
It is a relation but not a function because a single element in the domain has been mapped to two elements in the co domain. Isn’t it??
Please tell me if I’m correct or not.
It is really confusing.
An onto function exists if and only the co-domain is equal to the range that is every element in set A (the domain) is mapped to every element in set B (the range/codomain) i.e without leaving out any element. Irrespective of whether it is a one to one mapping or not. Therefore it is a function. Put simply, take set A as a set of sons and set B as a set of fathers, a function requires that every son has one father (which is normal) yet every father can have more than one son(which is also normal) so if one element in set A maps to more than one element in set B it is not a function (and we will need DNA test to know who really is the father) but if more than one element in set A maps to one element in set B it is still a function (the elements are just brothers). What and onto function requires is that every father has a son. Sorry if I made it a bit complicated I feel that if I continue I might make it worse just it study a bit more and from different sources, videos or books and you will understand it better.