F of Gof Xis 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?

**fof g of x**is 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) = x^{2}. We will find f(g(3)). For 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 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) = 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) = 3x^{2}+ 2 and g(x) = √1 - x.

g(f(x)) = g(3x^{2}+ 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 = aon 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 = aon 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(x^{2})) = cos(x^{2}) · d/dx(x^{2}) = cos(x^{2}) · 2x = 2x cos(x^{2}).

**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

## FAQs on F of G of x

### What is the Definition of F of G of x?

**F of Gof 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) = x^{2} 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) = x^{2}and g(x) = sin x, then (i) f(g(x)) = f(sin x) = (sin x)^{2}= sin^{2}x whereas (ii) g(f(x)) = g(x^{2}) = sin x^{2}.

### 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 XFrom 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 XFrom 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).