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Let f(x) = x + 8 and g(x) = x2 − 6x − 7. Find f(g(2)).

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 ) $:

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

\[ g( f( 3 ) ) \ = \ 1331 \ – \ 2 \]

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

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