The **aim of this problem** is to shed light on the very basic concept of **composite functions.**

An expression or formula describing a **mathematical relationship** between two or more variables is **called a function**. A **composite function** is a type of function that is a **cascade of two or more functions**. In simpler words, we can say that if there are **two functions** (for example) then a composite function is the function of **output of the other function.**

Let’s try to understand it with the **help of an example**. Let’s say that there are two functions, $ f $ and $ g $. Now the **composite function**, usually symbolized by $ fog $, is defined as follows:

\[ fog \ = \ f( g( x ) ) \]

This shows that to** obtain the function** $ fog $, we must use the **output of the function** $ g $ as the **input of function** $ f $.

## Expert Answer

**Given:**

\[ g( x ) \ = \ x^{ 2 } \ – \ 6x \ – \ 7 \]

**Substituting $ x \ = \ 2 $ in $ g( x ) $:**

\[ g( 2 ) \ = \ ( 2 )^{ 2 } \ – \ 6 ( 2 ) \ – \ 7 \]

\[ g( 2 ) \ = \ 4 \ – \ 12 \ – \ 7 \]

\[ g( 2 ) \ = \ 15 \]

**Given:**

\[ f( x ) \ = \ x \ + \ 8 \]

**Substituting $ x \ = \ g( 2 ) \ = 15 $ in $ f( x ) $:**

\[ f( g( 2 ) ) \ = \ 15 \ + \ 8 \]

\[ f( g( 2 ) ) \ = \ 23 \]

Which is the desired result.

## Numerical Result

\[ f( g( 2 ) ) \ = \ 23 \]

## Example

If $ f( x ) \ = \ x^{ 2 } \ + \ 2 $ and $ g( x ) \ = \ x^{ 3 } \ – \ 2 $. **Find** $ g ( f ( 3 ) ) $.

**Given:**

\[ f( x ) \ = \ x^{ 2 } \ + \ 2 \]

**Substituting $ x \ = \ 3 $ in $ f( x ) $:**

\[ f( 3 ) \ = \ ( 3 )^{ 2 } \ + \ 2 \]

\[ f( 3 ) \ = \ 9 \ + \ 2 \]

\[ f( 3 ) \ = \ 11 \]

**Given:**

\[ g( x ) \ = \ x^{ 3 } \ – \ 2 \]

**Substituting $ x \ = \ f( 3 ) \ = 11 $ in $ g( x ) $:**

\[ g( f( 3 ) ) \ = \ ( 11 )^{ 3 } \ – \ 2 \]

\[ g( f( 3 ) ) \ = \ 1331 \ – \ 2 \]

\[ g( f( 3 ) ) \ = \ 1329 \]