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Given the following functions, find f of g of h.

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

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:

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

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

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

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