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**Unformatted text preview: **MATH 2400 LECTURE NOTES: POLYNOMIAL AND RATIONAL FUNCTIONS PETE L. CLARK Contents 1. Polynomial Functions 1 2. Rational Functions 6 1. Polynomial Functions Using the basic operations of addition, subtraction, multiplication, division and composition of functions, we can combine very simple functions to build large and interesting (and useful!) classes of functions. For us, the two simplest kinds of functions are the following: Constant functions : for each a ∈ R there is a function C a : R → R such that for all x ∈ R , C a ( x ) = a . In other words, the output of the function does not depend on the input: whatever we put in, the same value a will come out. The graph of such a function is the horizontal line y = a . Such functions are called constant . The identity function I : R → R by I ( x ) = x . The graph of the identity function is the straight line y = x . Recall that the identity function is so-called because it is an identity element for the operation of function composition: that is, for any function f : R → R we have I ◦ f = f ◦ I = f . Example: Let m, b ∈ R , and consider the function L : R → R by x 7→ mx + b . Then L is built up out of constant functions and the identity function by addition and multiplication: L = C m · I + C b . Example: Let n ∈ Z + . The function m n : x 7→ x n is built up out of the iden- tity function by repreated multiplication: m n = I · I ··· I ( n I ’s altogether). The general name for a function f : R → R which is built up out of the identity function and the constant functions by finitely many additions and multiplications is a polynomial . In other words, every polynomial function is of the form (1) f : x 7→ a n x n + . . . + a 1 x + a for some constants a , . . . , a n ∈ R . 1 2 PETE L. CLARK However, we also want to take – at least until we prove it doesn’t make a dif- ference – a more algebraic approach to polynomials. Let us define a polynomial expression as an expression of the form ∑ n i =0 a i x i . Thus, to give a polynomial expression we need to give for each natural number i a constant a i , while requiring that all but finitely many of these constants are equal to zero: i.e., there exists some n ∈ N such that a i = 0 for all i > n . Then every polynomial expression f = ∑ n i =0 a i x i determines a polynomial func- tion x 7→ f ( x ). But it is at least conceivable that two different-looking polynomial expressions give rise to the same function . To give some rough idea of what I mean here, consider the two expressions f = 2 arcsin x + 2 arccos x and g = π . Now it turns out for all x ∈ [ − 1 , 1] (the common domain of the arcsin and arccos func- tions) we have f ( x ) = π . (The angle θ whose sine is x is complementary to the angle φ whose cosine is x , so arcsin x +arccos x = θ + φ = π 2 .) But still f and g are given by different “expressions”: if I ask you what the coeﬃcient of arcsin x is in the expression f , you will immediately tell me it is 2. If I ask you what the coeﬃ-, you will immediately tell me it is 2....

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