# 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 $f$.

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 $g( x )$:

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

$g( f( 3 ) ) \ = \ 1331 \ – \ 2$

$g( f( 3 ) ) \ = \ 1329$