Math 1010 - Lecture 19 NotesDylan ZwickFall 2009In this lecture we’ll define and discuss rational functions, and go overthe basic operations of multiplication and division. We’ll find that thingswork out pretty much the same way they do with our usual fractions weall know and love. The hardest thing is simplifying, as it will involvefactoring.1 Rational FunctionsA rational function is a function made up of one polynomial divided byanother. So, for example, the function f(x) below would be a rationalfunction:f(x) =x2+ 2x + 52x − 7.The domain of a rational function, which is the set of real numbersfor which the function is defined, will be a ll real numbers for which thedenominator is not zero. So, in our example the only number for whichour denominator is z ero is x = 7/2. So, the domain of the function f(x) isall real numbers not equa l to 7/2. We could write this as:(−∞, 7/2) ∪ (7/2, ∞),or just as x 6= 7/2.1ExamplesFind the domain of the following rational functions:1.3x − 37.2.x2+ 7x2− 4.3.3x + 11x2+ 4.Just as we can simplify fractions, if we have a common term in bothour numerator and denominator we can simplify rational functions. Thedifference is that with rational functions these common terms are polyno-mials.So, for example, if f(x), g(x), and h(x) are polynomials with g(x) 6= 0and h(x) 6= 0 then:f(x)h(x)g(x)h(x)=f(x)g(x).In other words, we can cancel the function h(x) the same way we cancela common multiple with fractions of integers. The only caveat here is the2points where h(x) = 0. Our old function will not be defined at these points,but our new function may be. So, to be correct, we require that our domainbe the same as the domain of our old function.For example:x2+ 4x + 4x2− 4=(x + 2)(x + 2)(x + 2)(x − 2)=x + 2x − 2.with the caveat the our domain remains x 6= ±2.ExamplesSimplify the following rational functions.1.2x − 34x − 6.2.x2− 7xx2− 14x + 49.33.x4− 25x2x2+ 2x − 15.2 Multiplying and Dividing Rational FunctionsWe multiply and divide rational functions in exactly the same way wemultiply and divide fractions.If f(x), g(x), h(x), w(x) are polynomials with g(x) 6= 0 and w(x) 6= 0then:f(x)g(x)·h(x)w(x)=f(x)h(x)g(x)w(x).And similarly for division, but here we require h(x) 6= 0 as well:f(x)g(x)÷h(x)w(x)=f(x)w(x)f(x)h(x).4ExamplesCalculate the following:1.(2x − 3)(x + 8)x3·x3 − 2x.2.2t2− t − 15t + 2·t2− t − 6t2− 6t + 9.3.x2+ 95(x + 2)÷x + 35(x2− 4).54.(x3y)2(x + 2y)2÷x2y(x +