\[ h(x) = (x + 2)^3 \]

The question aims to find the **functions** **f** and **g** from a **third function** which is a **composition** of the **function** of those two functions.

The **composition** of **functions** can be defined as putting one **function** into **another function** that **outputs** the **third function.** The **output** from one function goes as **input** to the other function.

## Expert Answer

We are given a **function h(x)** which is a **composition** of **functions** **f and g**. We need to find these **two functions** from **h(x).**

\[ (f \circ g) (x) = f( g(x) ) = h(x) = (x + 2)^3 \]

First we can assume the value of **g(x)** from the given **composition function** and then we can calculate the value of **f(x).** It can also be done **conversely** assuming the value of **f(x)** and then calculating **g(x).**

Lets assume **g(x)** and then find **f(x)** using **h(x).**

\[ Assuming\ g(x) = x + 2 \]

Then **f(x)** will be:

\[ f(x) = x^3 \]

Using these **function values,** if we calculate **h(x)** or $ (f \circ g) (x)$, it should give us the same **output function.**

\[ h(x) = f \circ g (x) = ( g(x) )^3 \]

\[ h(x) = (x + 2)^3 \]

We can also assume other values of **g(x)** and the respective **f(x)** are given as follows:

\[ g(x) = x \hspace{0.8in} f(x) = (x + 2)^3 \]

\[ g(x) = x + 1 \hspace{0.8in} f(x) = (x + 1)^3 \]

\[ g(x) = x\ -\ 1 \hspace{0.8in} f(x) = (x + 3)^3 \]

We can make a lot of different **combinations** for these **functions,** and they should give out the same **h(x).**

## Numerical Result

\[ f(x) = x^3 \hspace{0.6in} g(x) = x + 2 \]

\[ f(x) = (x + 2)^3 \hspace{0.6in} g(x) = x \]

\[ f(x) = (x + 1)^3 \hspace{0.6in} g(x) = x + 1 \]

## Example

Find the **functions** **f** and **g** such that $( g \circ f ) (x) = h(x)$.

\[ h(x) = x + 4 \]

First, we assume **f(x)** as the given **composition** of **functions** is $(g \circ f) (x)$.

\[ Assuming\ f(x) = x + 1 \]

The respective **g(x)** for this **f(x)** which satisfy the given **composition** of **functions** is:

\[ g(x) = x + 3 \]

We can verify it if it **satisfies** the **condition** we find $(g \circ f) (x)$ using the **functions** that we calculated.

\[ g(x) = x + 3 \]

\[ g( f(x) ) = ( x + 1 ) + 3 \]

\[ h(x) = x + 1 + 3 \]

\[ h(x) = (g \circ f) (x) = x + 4 \]

This is the same **composition** of **function** as given in the question statement, so we can conclude that the **functions** **f** and **g** that we calculated are **correct.**

There can also be other **functions f** and **g** that will satisfy the condition of giving out the same **composition** of **functions** $(g \circ f) (x)$. Here are some of the other **g and f functions** that are also correct.

\[ f(x) = x + 2 \hspace{0.6in} g(x) = x + 2 \]

\[ f(x) = x + 3 \hspace{0.6in} g(x) = x + 1 \]

\[ f(x) = x \hspace{0.6in} g(x) = x + 4 \]