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Find two functions f and g such that (f ∘ g)(x) = h(x).

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

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.

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

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

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

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