###### Answer: B. all real values of x except x=5

More formally f = g if f ( x ) = g ( x ) for all x ∈ X where f : X → Y and g : X → Y. [8] [9] [note 4] The domain and codomain are not always explicitly given when a function is defined and without some (possibly difficult) computation one might only know that the domain is contained in a larger set.

Inverse function – Wikipedia

Derivative – Wikipedia

Function composition – Wikipedia

Derivative – Wikipedia

Given the function symbols F and G one can introduce a new function symbol F ∘ G the composition of F and G satisfying ( F ∘ G )( X ) = F(G ( X )) for all X . Of course the right side of this equation doesn’t make sense in typed logic unless the domain type of F matches the codomain type of G so this is required for the composition to be defined.

Then f ( x ) g ( x ) = 4x 2 + 4x + 1 = 1. Thus deg( f ⋅ g ) = 0 which is not greater than the degrees of f and g (which each had degree 1). Since the norm function is not defined for the zero element of the ring we consider the degree of the polynomial f ( x ) = 0 to also be undefined so that it follows the rules of a norm in a Euclidean domain .

Let f be a function whose domain is the set X and whose codomain is the set Y.Then f is invertible if there exists a function g with domain Y and codomain X with the property: = ⇔ =. If f is invertible then the function g is unique which means that there is exactly one function g satisfying this property. Moreover it also follows that the ranges of g and f equal their respective codomains.

Sat Nov 08 2003 13:30:00 GMT-0500 (Eastern Standard Time) · However since F [ X ] is a unique factorization domain there is a unique representation for any rational expression P/Q with P and Q polynomials of lowest degree and Q chosen to be monic. This is similar to how a f…