Question #aee4a

1 Answer
Nov 4, 2017

#f(x)=x^4,\ g(x)=x-5, \ F(x)=f@g(x)=(x-5)^4#

Explanation:

Functions are a powerful tool for capturing and expressing mathematical patterns; they are defined by a set of rules which map an element or elements from one set to a corresponding element of another. The most common sort of functions you'll encounter in grade school algebra courses are ones which map some number from a set called the domain to a number in a set called the range of the function according to rules defined by some algebraic expression.

When examining the behavior of a more function, it can be useful to break it down into a composition of more elementary functions, which we can do by breaking down each of the "rules" being applied to the original number.

The function #F# takes some real number #x# and maps it to the value #F(x)# by the algebraic rule #(x-5)^4#. We can break this single rule into two separate rules:

  • First, #x# has #5# subtracted from it, producing the expression #x-5#
  • Next, #x-5# is raised to the #4#th power, giving us #(x-5)^4#

We can turn each of these rules into its own function; we'll define a function #g# by the rule #x-5# and a function #f# by the the rule #x^4#, or in full notation:

#g(x)=x-5#
#f(x)=x^4#

We can then define #F# by "feeding" the function #g(x)# through the function #f(x)#, obtaining #f(g(x))=(x-5)^4#. We call the function #f(g(x))# the composition of the functions #f# and #g#, and we denote that function as #f@g(x)#. So, we can define the function #F# in terms of this composition as:

#F(x)=f@g(x)=(x-5)^4#