Two Functions F and G Neither of Which is Continuous at 0

F of G of x

F of G of X is a composite function made of two functions f(x) an g(x). Let us understand f of g of x by a real-life example. In the process of preparing french fries, we use the slicer and the fryer. Let us assume that x is the potato, the slicer is doing the function g(x) (which is slicing the potato) and the fryer is doing the function f(x) (frying the potato). Then f of g of x represents the process of preparing french fries because:

• First slice the potato - it means find g(x).
• Then use the sliced potatoes in the fryer - i.e., use g(x) in f(x), which gives f of g of x.

Let us learn more about f of g of x along with its mathematical definition, domain, range, and how to find it in different scenarios.

1.	What is F of G of x?
2.	How to Find F of G of x?
3.	Finding F of G of x From Graph
4.	Finding F of G of x From Table
5.	Domain and Range of F of G of x
6.	Derivative of F of G of x
7.	FAQs on F of G of x

What is F of G of x?

f of g of xis also known as a composite function and it is mathematically denoted as f(g(x)) or (f ∘ g)(x) and it means that x = g(x) should be substituted in f(x). It is also read as "f circle g of x". It is an operation that combines two functions to form another new function. In f of g of x, the output of one function becomes the input of the other function. It can be thought of as a series of machines or operations.

Symbol of f of g of x

The symbol of a composite function is '∘'. Sometimes it is represented by just using the brackets without using the symbols. For any two functions f and g, there can be two composite functions:

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

How to Find F of G of x?

We know that whenever we are simplifying some mathematical expression, we first operate the things that are inside the brackets. So for finding f(g(x)), we have to first find g(x) and then take g(x) as input of f(x) and simplify. Here is an example to understand this. Let us assume that f(x) = 2x + 3 and g(x) = x². We will find f(g(3)). For this:

• Step 1: Find g(3).
  g(3) = 3² = 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 visualize this process using the following figure easily.

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).

More Examples of F of G of x

Here are more examples of finding f(g(x)).

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

Finding F of G of x From Graph

Sometimes f and g are not defined algebraically. Instead, the graphs of f and g are given and we will be asked to find f(g(x)). To find f(g(x)) from graph for some number x = a:

• Find g(a) by using the graph of g(x) (see the corresponding y-value of x = a on the graph of g)
• Find f(g(a)) by using the graph of f(x) (see the corresponding y-value of x = g(a) on the graph of f)

Here is an example.

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.

Finding F of G of x From Table

Sometimes f and g are defined by a table representing each function. In that case, to find f(g(x)) at some number x = a:

• Find g(a) by using the table of g(x) (see the corresponding y-value of x = a on the table of g)
• Find f(g(a)) by using the table of f(x) (see the corresponding y-value of x = g(a) on the table of f)

Here is an example.

Example: Find f(g(-7)) from the following tables.

x	f(x)
-3	15
-5	19
-7	23
-9	27
-11	31

x	g(x)
-3	-7
-5	-9
-7	-11
-9	-13

Solution:

Let us find f(g(-7)).

f(g(-7)) = f(-11) (∵ (-7, -11) lies on g ⇒ g(-7) = -11)

= 31 (∵ (-11, 31) lies on f ⇒ f(-11) = 31)

Therefore, f(g(-7)) = 31.

Domain and Range of F of G of x

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. Then 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)).

Domain of F of G of x

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

• Step 1:Find the domain of g(x) and denote it by A.
• Step 2:Find the domain of the resultant function f(g(x)) and denote it by B.
• Step 3:Find their intersection (A ∩ B) which gives the domain of f(g(x))

Example: Find the domain of f(g(x)) when f(x) = 2/(x - 1) and g(x) = 3/(x - 2).

Solution:

First, we will find f(g(x)).

\(\begin{aligned}
f(g(x)) &=f\left(\frac{3}{x-2}\right) \\[0.2cm]
&=\frac{2}{\frac{3}{x-2}-1} \\[0.2cm]
&=\frac{2}{\frac{3-x+2}{x-2}} \\[0.2cm]
&=\frac{2(x-2)}{5-x}
\end{aligned}\)

Finding domain of inner function g(x):
Since g(x) = 3/(x- 2), it is NOT defined at x = 2.
So the domain of g(x) is {x : x ≠ 2}

Finding domain of resultant function f(g(x)):
Since f(g(x)) = \(\frac{2(x-2)}{5-x}\), it is NOT defined at x = 5.
So the domain of resultant function is { x : x ≠ 5}

Now, the domain of f(g(x))
= {x : x ≠ 2} ∩ { x : x ≠ 5}
= (-∞, 2) U (2, 5) (2, ∞)

Note: Though f(g(x)) = \(\frac{2(x-2)}{5-x}\) is defined at x = 2, 2 is NOT present in the domain of f(g(x)) because g(x) is NOT defined at x = 2. So for f(g(x)) to exist at some x value, g(x) should exist first at at that x value.

Range of F od G of x

The range of f of g of x does not depend upon the inner function. So we just compute the range of f(g(x)) using the techniques of finding the range of a function.

Derivative of F of G of x

In Calculus, we find the derivative of a composite function, f(g(x)) using the chain rule. The chain rule says:

• d/dx (f(g(x)) = f '(g(x)) · g'(x)

Here is an example.

d/dx (sin(x²)) = cos(x²) · d/dx(x²) = cos(x²) · 2x = 2x cos(x²).

Important Points on F of G of x:

• f of g of x is a composite function that is represented by f(g(x)) (or) (f ∘ g)(x).
• To find f(g(x)), substitute g(x) into f(x).
• To find the domain of f(g(x)), find the domain of both the inner function g(x) and the resultant function f(g(x)) and then compute the intersection.
• To find the range of f(g(x)), use the usual techniques of finding the range of a function.

☛Related Topics:

• Linear Function Calculator
• Quadratic Function Calculator
• Graphing Functions Calculator
• Inverse Function Calculator

Examples on F of G of x

1. Example 1: Find f(g(x)) when f(x) = √x + 3 and g(x) = 5 - x

   Solution:

   We can find f of g of x (f(g(x)) by substituting g(x) into f(x).

   f(g(x)) = f(5 - x)

   = √5 - x + 3

   = √-x + 8

   Answer: f(g(x)) = √-x + 8.

2. Example 2: Find the domain of f(g(x)) with respect to the functions from Example 1.

   Solution:

   Finding the domain of the inner function (g(x)):

   Since g(x) = 5 - x is a polynomial, it defined everywhere,
   the domain of inner function = the set of all real numbers = R

   Finding the domain of the resultant function f(g(x)):

   From Example 1, f(g(x)) = √-x + 8. To find its domain:
   -x + 8 ≥ 0
   8 ≥ x (or) x ≤ 8
   So the domain of the resultant function = {x : x ≤ 8}

   Now, the domain of f of g of x OR f(g(x)) = R ∩ {x : x ≤ 8} = {x : x ≤ 8}

   Answer: The domain of f(g(x)) = (-∞, 8]

3. Example 3: Find f and g such that the function h(x) = sin (x³ + 2) is a composite function f of g of x. Also, verify your answer.

   Solution:

   To decompose a function as a composite function (f of g of x) of two functions, remember to define the inside function to be g(x) and the outside function to be f(x).

   So here, f(x) = sin x and g(x) = x³ + 2.

   Verification:

   f(g(x)) = f(x³ + 2) = sin(x³ + 2).

   Answer: f(x) = sin x and g(x) = x³ + 2.

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Practice Questions on F of G of x

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FAQs on F of G of x

What is the Definition of F of G of x?

F of G of X is written as f(g(x)) and it is called a composite function. It is obtained by replacing x in f(x) with g(x).

What is the Process of Finding F of G of x?

To find f(g(x)), we just substitute x = g(x) in the function f(x). For example, when f(x) = x² and g(x) = 3x - 5, then f(g(x)) = f(3x - 5) = (3x - 5)².

What is the Difference Between F of G of x and G of F of x?

"f of g of x" is written as f(g(x)) and "g of f of x" is written as g(f(x)).

• f(g(x)) = a function obtained by replacing x with g(x) in f(x).
• g(f(x)) = a function obtained by replacing x with f(x) in g(x).

For example, if f(x) = x² and g(x) = sin x, then (i) f(g(x)) = f(sin x) = (sin x)² = sin² x whereas (ii) g(f(x)) = g(x²) = sin x².

How to Find the Domain of F of G of x?

To find the domain of f(g(x)):

• Find the domain of inner function g(x).
• Find f(g(x)) algebraically and find its domain.
• Find the intersection of both domains.

How to Find the Range of F of G of x?

The range of f(g(x)) doesn't depend on either the range of f or the range of g. So the range of f(g(x)) is found just like the range of any other function.

How to Find F of G of X From a Table?

To find the value of f(g(x)) for some x = k:

• Find g(k) from the table of g.
• Find f(g(k)) from the table of f.

How to Find F of G of X From a Graph?

To find the value of f(g(x)) for some x = k:

• Find g(k) from the graph of g.
• Find f(g(k)) from the graph of f.

How to Find the Derivative of F of G of X?

f of g of x is a composite function and so the chain rule of differentiation is used to find its derivative. This rule says, d/dx (f(g(x)) = f'(g(x)) × g'(x).

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Source: https://www.cuemath.com/calculus/f-of-g-of-x/

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