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The f of g of x is also recognized as a composite function and is signified mathematically as f(g(x)) or (f ∘ g)(x). It is a process that combines two functions to create a new function. The output of one function becomes the input of the other function in f of g of x. It can be viewed as a collection of machines or operations. Let us look at a real-world example to better understand f of g of x. The slicer and fryer are used in the preparation of french fries. Presume x is the potato, the slicer is performing the function g(x) (slicing the potato), and the fryer is performing the function f(x) (frying the potato). The procedure of preparing french fries is represented by f of g of x because:

First, slice the potato - this means locate g(x).

Then, in the fryer, use the sliced potatoes - i.e., g(x) in f(x), which gives f of g of x.

A composite function is represented by the symbol '∘'. Its often depicted by just using the brackets and not the symbols. There can be two composite functions for any two functions, f, and g:

- f of g of x = (f ∘g)(x) = f(g(x))
- g of f of x = (g ∘f)(x) = g(f(x))

When we ease a mathematical expression, we all know that we start with the things inside the brackets. So, in order to find f(g(x)), we must first find g(x), then use g(x) as an input to f(x) and simplify. Here is an instance to help people understand. Let's say f(x) = 2x + 3 and g(x) = x^{2}. We will discover f(g(3)). Because of this:

Step 1: Find g(3).

g(3) = 3^{2} = 9.

Step 2: Find f(g(3)) by using g(3) as input for f(x).

f(g(3)) = f(9) = 2(9) + 3 = 18 + 3 = 21.

We can easily visualise this procedure by using the figure below.

Thus:

To find f(g(x)), substitute x = g(x) into f(x).

To find g(f(x)), substitute x = f(x) into g(x).

Here are some instances of finding f(g(x)).

Example 1: Find f(g(x)) when f(x) = 3x^{2} + 2 and g(x) = √1 - x.

f(g(x)) = f(√1 - x)

= 3(√1 - x)^{2} + 2

= 3(1 - x) + 2

= 3 - 3x + 2

= 5 - 3x

Example 2: Find g(f(x)) when f(x) = 3x2 + 2 and g(x) = √1 - x.

g(f(x)) = g(3x^{2} + 2)

= √1 - (3x² + 2)

= √1 - 3x² - 2.

= √-3x² - 1.

Occasionally f and g are not described algebraically. As an alternative, the graphs of f and g are given, and we will be asked to discover f(g(x)). To discover f(g(x)) from graph for certain number x = a:

- Discover g(a) by using the graph of g(x) (see the corresponding y-value of 'a' on the graph of g); and
- Discover f(g(a)) by using the graph of f(x) (see the corresponding y-value of 'g(a)' on the graph of f).

Here is an instance.

Example: Find f(g(-2)) from the following graph.

Solution:

Let us find f(g(-2)) from the above graph.

f(g(-2)) = f(2) (∵ (-2, 2) lies on g ⇒ g(-2) = 2)

= 4 (∵ (2, 4) lies on f ⇒ f(2) = 4)

Thus, f(g(-2)) = 4.

The domain of a function y = f(x) is the set of all x values where it is defined (i.e., it is the set of all inputs), and the range is the set of all y-values that the function produces (i.e., it is the set of all outputs). In general, if a function g: A → B and f: B → C, then f of g of x is a function such that f ∘ g: A → C. The domain of f ∘ g is A, and the range of f ∘ g is C. But it cannot be the case all the time. Let us see how to find the domain and range of f(g(x)).

**Ans: **The function fg(x) is the product of the functions f and g, while the function f(g(x)) is the composition of the function f and g.

**Ans: **The domain of a composite function depends not only on the resultant function but also on the inner function. To discover the domain of f(g(x)):

- Discover the domain of g(x) and denote it by A;
- Discover the domain of the resultant function f(g(x)) and denote it by B; and
- Discover their intersection (A ∩ B), which gives the domain of f(g(x))

**Ans: **The inner function does not affect the range of f of g of x. Thus, we compute the f(g(x)) range using function range computation techniques.

**Ans: **Occasionally f and g are described by a table representing each function. In that case, to discover f(g(x)) at some number x = a:

- Discover g(a) by using the table of g(x) (see the corresponding y-value of 'a' on the table of g); and
- Discover f(g(a)) by using the table of f(x) (see the corresponding y-value of 'g(a)' on the table of f).

**Ans: **Occasionally f and g are not described algebraically. As an alternative, the graphs of f and g are given, and we will be asked to discover f(g(x)). To discover f(g(x)) from graph for certain number x = a:

- Discover g(a) by using the graph of g(x) (see the corresponding y-value of 'a' on the graph of g); and
- Discover f(g(a)) by using the graph of f(x) (see the corresponding y-value of 'g(a)' on the graph of f).

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