\[ \left \{ \begin{array}{ l } f( x ) \ = \ x^{ 2 } \ + \ 1 \\ g( x ) \ = \ 2 x \\ h( x ) \ = \ x \ – \ 1 \end{array} \right. \]

This **question aims** to explain and apply the key concept of **composite functions** used in fundamental algebra.

An **algebraic function** can be defined as a **mathematical expression** that describes or **models the relationship** between two or more variables. This expression must have a **one to one mapping** between input and output variables.

If we build such a system that the output of **one function is used as the input of the other function**, then such a **cascade or causal** relationship between two variables and some intermediate variables is called a **composite function.** In simpler words, if the** input of a function** is the **output of some other function** than such a function may be called a **composite function**. For** example, **lets say that we are given with **two functions** denoted as $ f $ and $ g $. In this case the **composite function**, conventionally **symbolized** by $ fog $ or $ g0f $ may be defined by the following expression:

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

This shows that if we wish to** evaluate the function** $ fog $, we must have to use the **output of the first function** $ g $ as the **input of the second function** $ f $.

## Expert Answer

**Given:**

\[ \left \{ \begin{array}{ l } f( x ) \ = \ x^{ 2 } \ + \ 1 \\ g( x ) \ = \ 2 x \\ h( x ) \ = \ x \ – \ 1 \end{array} \right. \]

**Substituting $ x \ = \ h( x ) \ = \ x \ – \ 1 $ in $ g ( x ) $:**

\[ goh \ = \ g ( h ( x ) ) \ = \ 2 ( x \ – \ 1 ) \]

\[ goh \ = \ g ( h ( x ) ) \ = \ 2 x \ – \ 2 \]

**Substituting $ x \ = \ goh \ = \ 2 x \ – \ 2 $ in $ f ( x ) $:**

\[ fogoh \ = \ f ( g ( h ( x ) ) ) \ = \ ( 2 x \ – \ 2 )^{ 2 } \ + \ 1 \]

\[ fogoh \ = \ f ( g ( h ( x ) ) ) \ = \ ( 2 x )^2 \ + \ ( 2 )^2 \ – \ 2 ( 2 x ) ( 2 ) \ + \ 1 \]

\[ fogoh \ = \ f ( g ( h ( x ) ) ) \ = \ 4 x^2 \ + \ 4 \ – \ 8 x \ + \ 1 \]

\[ fogoh \ = \ f ( g ( h ( x ) ) ) \ = \ 4 x^2 \ – \ 8 x \ + \ 5 \]

Which is the desired result.

## Numerical Result

\[ fogoh \ = \ f ( g ( h ( x ) ) ) \ = \ 4 x^2 \ – \ 8 x \ + \ 5 \]

## Example

Find the value of the above composite function at x = 2.

**Recall:**

\[ fogoh \ = \ f ( g ( h ( x ) ) ) \ = \ 4 x^2 \ – \ 8 x \ + \ 5 \]

**Substituting x = 2 in above equation:**

\[ fogoh \ = \ f ( g ( h ( 2 ) ) ) \ = \ 4 ( 2 )^2 \ – \ 8 ( 2 ) \ + \ 5 \]

\[ fogoh \ = \ f ( g ( h ( 2 ) ) ) \ = \ 16 \ – \ 16 \ + \ 5 \]

\[ fogoh \ = \ f ( g ( h ( 2 ) ) ) \ = \ 5 \]