Overview of Composite Function

It is the process of integrating two or more functions into a single function that is referred to as a composition of functions. A function is a representation of some kind of effort. Take, for example, the process of making bread. Let x represent the flour, and let g(x) represent the function of the food processor performing the function of preparing the dough using the flour, and let f(x) represent the function of the oven performing the function of baking the bread. Let x represent the flour. In order to make bread, the output of the function g(x) needs to be sent through the function f(x) (i.e., the prepared dough should be placed in the oven). The outcome is indicated by the symbol f(g(x)), and it is a composition of the functions f(x) and g(x) (x).

Let us look at what the composition of functions in mathematics is, as well as how to calculate it. Let’s look at how to determine its domain and range as well.

Composition of Functions

In the case of two functions f(x) and g(x), the composition is represented by f(g(x)) or (f o g) (x).It is a function that combines two or more functions to produce another function. When two functions are combined, the output of one function that is included within the parentheses becomes the input of the other function that is outside the parenthesis. i.e.,

• In the expression f(g(x)), g(x) is the input of f(x)
• In the expression g(f(x)), f(x) is the input of g(x)

Process of Solving Composite Functions 

When using BODMAS, we always begin by simplifying whatever is contained within the brackets. So, in order to determine f(g(x)), it is necessary to first compute g(x), which will then be substituted within f. (x). In the same way, in order to determine g(f(x)), it is necessary to first compute f(x) and then insert it into g. (x). In other words, the sequence in which the composite functions are discovered is important. It follows that f(g(x)) may or may not be equivalent to g(f(x)). We can obtain the composite function f(g(a)) for any pair of functions f(x) and g(x) by following the methods outlined below:

Find g(a) by replacing x = an in the g expression (x).

Find f(g(a)) by inserting x = g(a) in f and solving for f. (x).

We can better comprehend these stages if we look at the following example. We are looking for f(g-1) in this case when f(x) = x2 – 2x and g(x) = x – 5.

Domain of Composite Functions

In general, if g: X →Y and f: Y→Z, then f g: X →Z is equal to 1. In other words, the domain of fog is X and the range of f o g is Z. However, if the functions are specified algebraically, the following are the steps to take in order to determine the domain of the composite function f(g(x)): 1.

Figure out what the domain of the inner function g is (x) (Let’s say the letter A.)

Figure out what the domain of the function is that you obtained by solving for f(g(x)). (It should be B.)

Find the intersection of A and B, and the intersection of A and B provides the domain of f(g(x)).

Range of Composite Functions

The range of a composite function is computed in the same way that the range of any other function is computed. It is not dependent on the interior or outward functions of the body. Let us now compute the range of f(g(x)) that was demonstrated in the previous example. We get the expression f(g(x)) = (x + 3)/(2 x + 7). Assume that y = (x + 3)/(2 x + 7) is true. This is an example of a logical function. As a result, we solve it for x and set the denominator to be greater than zero in order to obtain the range.

(2x + 7) y =  x + 3

2xy +7y =x + 3

2xy – x = 3 – 7y

x (2y – 1) = 3 – 7y

x = (3 – 7y) / (2y – 1)

To find the range, 2y – 1 ≠ 0 which gives y ≠ 1/2.

As a result, the range is defined as {y : y ≠ 1/2}.

Conclusion

It is the process of integrating two or more functions into a single function that is referred to as a composition of functions. A function is a representation of some kind of effort.In the case of two functions f(x) and g(x), the composition is represented by f(g(x)) or (f o g) (x).It is a function that combines two or more functions to produce another function. When two functions are combined, the output of one function that is included within the parentheses becomes the input of the other function that is outside the parentheses. The range of a composite function is computed in the same way that the range of any other function is computed. It is not dependent on the interior or outward functions of the body.