Show that a set of real rational functions is a field.

NOTE: In this description, R represents the symbol for the set of real numbers. I couldn't find a way to type or copy the correct R symbol for the set of real numbers. Also, the parentheses in R(x) is used to distinguish the ring R(x) of rational functions from the ring R[x] of polynomials.

Show that the set R(x) of rational functions p(x)/q(x), where p(x), q(x) are in R[x] and q(x) <> 0, is a field.

Solution Summary

It is shown that a set of real rational functions is a field. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

See the attached file.
Consider the set of integers Z. The set Z x Z consists of all ordered pairs of integers. In symbols,
Z x Z={(x,y):x,y∈Z}
For example, (2,-5) , (0,0) , (-127,10) , and (-5,2) are all distinct elements of Z x Z.
Notice that the order matters; (2,-5) and (-5,2) are different elements of Z x Z.

A- Showthat if a function is continuous on all of R and equal to 0 at every rational point then it must be identically 0 on all of R
b- if f and g are continuous on all of R and f(r)=g(r) at every rational point,must f and g be the same function?

1) Rationalfunctions, graph and show asymptotes.
a) r(x)=1/x-4
b) r(x)=2x/1-x^2
c) r(x)= x^3+1/x^2-1
2) Define the inverse trigonometric functions for sinx & cosx.

Define then;
Let W = the set of whole numbers
F = the set of (non-negative) Fractions
I = the set of integers
Q= the set of rational numbers
R =the set of real numbers
Question
List all of the sets that have the following properties.
(a) 5 is an element of the set(s)?
(b) -1/2 is an e

I do not understand how to conduct these equations I Excel so that I can show a graph of each. Choose a second-order/third-order (e.g., x2/x3) and a third-order/second-order (e.g., x3/x2) rational function. Provide a graph for the second-order rational function (e.g., x2), choosing x values in the range from -10 through +10.