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 4th 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