# Composition of Functions

# Definition

Given functions*f*and

*g*, the composition of

*f*with

*g*is denoted by

*f*, is the function defined by

_{°}g( *f* _{°} *g* ) ( *x* ) = *f* ( *g* ( *x* ) )

*g*with

*f*is denoted by

*g*, is the function defined by

_{°}f( *g* _{°} *f* ) ( *x* ) = *g* ( *f* ( *x* ) )

**NOTE:**

( *f* _{°} *g* ) ( *x* ) ( *g* _{°} *f* ) ( *x* )

# Practice Problems

**Given**

*Example 1.**f*(

*x*) =

*x*

^{2}+ 1, and

*g*(

*x*) = 1 -

*x*, find the following:

**(a)**(

*f*

_{°}

*g*) (

*x*)

**(b)**(

*g*

_{°}

*f*) (

*x*)

*Solution***(a)**(

*f*

_{°}

*g*) (

*x*)

(

*f*

_{°}

*g*) (

*x*) =

*f*(

*g*(

*x*) )

=

*f*( 1 -

*x*)

= ( 1 -

*x*)

^{2}+ 1

= 1 - 2

*x*+

*x*

^{2}+ 1

=

*x*

^{2}- 2

*x*+ 2

*Answer***(b)**(

*g*

_{°}

*f*) (

*x*)

*Solution*(

*g*

_{°}

*f*) (

*x*)

(

*g*

_{°}

*f*) (

*x*) =

*g*(

*f*(

*x*) )

=

*g*(

*x*

^{2}+ 1 )

= 1 -

*x*

^{2}+ 1

= 2 -

*x*

^{2}

*Answer***If**

*Example 2.**f*(

*x*) = ln ( log

*x*) and

*g*(

*x*) = 10

^{e-7}Then (

*f*

_{°}

*g*) (

*x*)=?

*Solution*(

*f*

_{°}

*g*) (

*x*) =

*f*(

*g*(

*x*) )

=

*f*( 10

^{e-7})

= ln ( log 10

^{e-7})

= ln

*e*

^{-7}

= -7

*Answer***NOTE:**Here we use the following facts:

log 10 = 1 and

ln

*e*= 1