Question

Question #aee4a

Answer 1

`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`