How to Solve a Composite Function Problem?

If we are given two functions, we can combine them to form a third function by composing the first function into the second function. The methods required to carry out this operation are comparable to those required to solve any function for any given value in any other situation. Composite functions are used to describe this type of function.

The process of composing a function is accomplished by substituting one function for another function.

For example, the composite function combining f (x) and g (x) is denoted by the symbol f [g (x)] (x). As a rule of thumb, the composite function f [g (x)] should be understood as “f of g of x.” In mathematics, the function g (x) is referred to as an inner function, whereas the function f (x) is referred to as an outer function. Because of this, the expression f [g (x)] can be taken to mean, “the function g is an inner function of the outer function f.”

Function Compositions Have Certain Characteristics

If there are three functions f, g, and h, and they are all associative, then they are said to be associative if and only if they satisfy the associative property of function composition.

f ∘ (g ∘ h) = (f ∘ g) ∘ h

In mathematics, the commutative property states that two functions f and g are commutent with each other if and only if

g ∘ f = f ∘ g

The following are some other characteristics:

•The function composition of a one-to-one function is always one to one, regardless of the function type.

•The function composition of two onto functions is always a function composition of two onto functions.

•As an example, consider the inverse of the composition of two functions f and g, which is the same as the composition of the inverse of both of the functions, as shown by  (f ∘ g)-1 = ( g-1 ∘ f-1).

How to Solve Composite Functions (with Examples)

In mathematics, solving a composite function refers to obtaining the result of the combination of two functions. For the composition of a function, a tiny circle (o) is used to symbolise its composition.

Follow the methods outlined below to gain an understanding of how to solve the given composite function.

Step 1: Write the given composition in a different style than you normally would.

Consider the functions f(x) = x2 and g(x) = 3x.

Now,

The expression (f g) (x) can be represented as f[g(x)].

Step 2: Using the individual functions as a reference, replace the variable x that is present in the outside function with the variable x that is present in the inside function.

That is to say,

Given the fact that (f ∘ g)(x) = f(3x) {since g(x) = 3x}.

Step 3: Finally, simplify the function that has been obtained.

(f ∘ g)(x) = f(3x) = (3x)2 {since f(x) = x2}

9×2

Composition of a Function in Relation to Itself

It is possible to combine two functions into a single one. For example, consider the function f. Then the composition of function f with itself will be

(f∘f)(x) = f(f(x))

Consider the following illustration to better grasp what we’re talking about:

Consider the following example: If f(x) = 3×2, then find (ff) (x).

Given,

f(x) = 3×2

(f∘f)(x) = f(f(x))

= f (3×2)

= 3(3x)2

= 3.9×2

= 27×2

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 a mathematical action known as the composition of a function, which is defined as an operation in which two functions, let us say f and g, combine to form one new function, let us call it h, such that the new function h(x) = g(f(x)).For example, the composite function combining f (x) and g (x) is denoted by the symbol f [g (x)] (x). As a rule of thumb, the composite function f [g (x)] should be understood as “f of g of x.”In mathematics, solving a composite function refers to obtaining the result of the combination of two functions. For the composition of a function, a tiny circle (o) is used to symbolise its composition.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.