Rational functions Definition and 4 Discussions

In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field K. In this case, one speaks of a rational function and a rational fraction over K. The values of the variables may be taken in any field L containing K. Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is L.
The set of rational functions over a field K is a field, the field of fractions of the ring of the polynomial functions over K.

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1. Possible values an expression can take : ##\dfrac{x^2-x-6}{x-3}##

Problem statement : Let me copy and paste the problem as it appears in the text : Attempt 1 (from text) : The book and me independently could solve this problem. I copy and paste the solution from the book below. Attempt 2 (my own) : The problem should afford a solution using the...
2. Question about asymptotes of rational function

I tried graphing the function in the calculator, and the graph seems to have a horizontal asymptote at y=0, not at y=1. Why is this so? Thanks for helping out.
3. Rational Equation Problem

Homework Statement Homework Equations y = f(x) y=k(x+4)(x)(x-6) y=1/f(x) y= 1/ (k(x+4)(x)(x-6)) The Attempt at a Solution I'm more looking for clarification on how people would approach this. There is no explicit point given to deduce the value of k to determine the vertical stretch or...
4. How to determine a hole in a graph?

I'm aware that in order to find the hole in a graph, you need to factor both the numerator and denominator, and look for terms that cancel out. However, is it merely just looking for a term that cancels out, or is it more specifically a term that cancels out and makes the numerator equal to...