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Relations and Functions

Composition of Functions and Invertible Function

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”.

composite functions

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)}.

composite functions

composite functions

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.

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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.

composite functions

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

                                                                     Learn Types of Functions here in detail. 

 Download Relations Cheat Sheet PDF by clicking on Download button below

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relation cheat sheet

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.

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