Composition of Functions and Invertible Function

A composite function is made up of multiple functions, with one function’s output feeding into the other. To put it another way, when the value of a function is determined by applying one function to an independent variable and the other to the result of another function whose domain consists of the values of the independent variable for which the first function’s result falls within the domain of the second. 

Composition of functions is the process of merging functions so that the output of one becomes the input of another. A composite function is the product of this process. The following notation is used to denote this combination:

(fg)(x)=f(g(x))

The left-hand side is written as “f” composed with g at x, and the right-hand side is written as “f of g of x “. The two sides of the equation are mathematically equal and have the same meaning. The composition operator is represented by the open circle symbol “”. 

Composition is a binary operation that combines two functions to create a new one, similar to how addition and multiplication combine two numbers to create a new one.

Invertible Function

If we reverse the order of mapping, we obtain the input as the new output, then the function is invertible. To put it another way, if a function f has a domain in set A and an image in set B, then f-1 has a domain in B and an image in A.

If an inverse function, denoted by the symbol f-1(x), consistently reverses the f(x) process, it is said to be the inverse function of f(x). 

If f(x) transforms a into b, then f-1(x) must transform b into a. 

In a more simple and formal sense, f-1(x) is the inverse function of f(x) if:

f(f-1(x))=x

We may use the supplied function to display the graph and check for invertibility to see if the function is invertible or not.

Properties of Composition of Function

• Composite functions are not commutative, i.e., (fg)(gf)

• If function f:AB and g:BC is One One function then (gf):AC is also One One function.

• The Composition of Function are associative, i.e., (fg)h=f(gh)

• If function f:AB and g:BC is Onto function, then (gf):AC is also Onto function.

Properties of Invertible Function

• Only one-to-one functions have an inverse. If g is the reciprocal of f, then f is the reciprocal of g.

• Both f and g are one-to-one functions if they are inverses of each other.

• If (fg)(x)=x, x is the domain of g, then f and g are inverses of each other.

• The range of f equals the domain of g, and the domain of f equals the range of g.

Conditions for Invertibility of the Function

To show that the function is invertible, we must show that it is both One to One and Onto, or Bijective.

When every domain element has a single image with codomain after mapping, we can say the function is One to One. When the function’s Range is equal to the codomain, the function is said to be Onto. When we show that a function is both One to One and Onto, we can say it is invertible.

Composition of Functions and Invertible Function Example

Example:

If f(x)=2×2-3x+4 and g(x)=5x-2, then calculate

1. (fg)(x)

2. (gf)(x)

Solution: 

1. Substituting g into f

(fg)(x)=f(g(x))=f(5x-2)=2(5x-2)2-3(5x-2)+4

=2(25×2-20x+4)-15x+6+4=50×2-40x+8-15x+10

=50×2-55x+18

2. Substituting f into g

(gf)(x)=g(f(x))=g(2×2-3x+4)=5(2×2-3x+4)-2

=10×2-15x+20-2=10×2-15x+18

Example: Let A:R-{4} and B:R-{1}. Consider the f:AB function, which has the formula f(x)=(x-3)/(x-4). Demonstrate that the function f(x) is invertible.

Solution: To demonstrate that the function is invertible, we must check the condition that the function is invertible, which we discussed earlier. To demonstrate that the function is invertible, we must first determine whether the function is One to One or not, therefore let’s do that.

Let x, yA such that f(x)=f(y)

(x-3)/(x-4)=(y-3)/(y-4)

(x-3)(y-4)=(y-3)(x-4)

xy-4x-3y+12=yx-4y-3x+12

-4x-3y=-4y-3x

4x+3y=4y+3x

x=y

This proves, f(x)=f(y), ∀x, yA, which tells that function is One to One.

Let’s look for Onto now. To demonstrate that f(x) is onto, we establish that its range equals its codomain.

Let y=(x-3)/(x-4)

Putting, f(x)=y

(x-3)/(x-4)=y

x-3=yx-4y

4y-3=yx-x

4y-3=x(y-1)

(4y-3)/(y-1)=x

Because xR-{4} and ∀yR-{1}, the range of f is given as =R-{1}. f=R-{1} is also a codomain.

Range=Codomainf is the Onto function.

Because both conditions are met, the function is One to One and Onto, making it Invertible.

Conclusion

In this article we learned about Composition of Function and Invertible Function. We started with the definition of Composition of Functions and Invertible Function then we found out about their properties and understood the concept with the help of Composition of Functions and Invertible Function examples. We also learned about the conditions for invisibility of function and how we can find the inverse of a function if it fulfils all the conditions.