# A Century of mathematics in America by Peter L Duren; Richard Askey; Uta C Merzbach; Harold M

By Peter L Duren; Richard Askey; Uta C Merzbach; Harold M Edwards

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Using (i)]. λ λ λ √ √ 1 1 1 = 1. [Using (ii)]. λ+ √ λ− √ = λ− (iv) λ λ λ (v) fn + fn+1 −1 λ n = 1 √ 5 λn − = 1 √ 5 λn + λn+1 − 1 √ 5 1 √ 5 1 √ 5 λn+2 − = 1 √ 5 λn+2 − = fn+2 . = = = λn+2 − λn+2 − (vi) f1 = 1, f2 = 1, f3 = 2 . . 1 +√ 5 −1 λ −1 λ −1 λ −1 λ n −1 λ n+2 n − n −1 λ −1 λ n+1 1 [using (i)] λ λ−1 λ 1 [using (ii)] λ2 1− n λn+1 − n+1 CHAPTER 2 Functions Aims and Objectives By the end of this chapter you will have • been introduced to formal terminology for sets and intervals; • deﬁned what is meant by a function; • revised polynomial, rational, and trigonometric functions; • been reminded how functions can be combined; • seen when and how functions have inverses.

The idea is to rewrite the quadratic as a sum of squared linear terms and a constant. We will consider ﬁrst a quadratic in x. We use the fact that (x + a)2 = x2 + 2ax + a2 so that x2 + 2ax = (x + a)2 − a2 . Thus x2 + 6x + 10 = (x + 3)2 − 9 + 10 = (x + 3)2 + 1. 12 Guide to Mathematical Methods If the leading coeﬃcient is not 1 then we take it out as a factor ﬁrst. 2x2 + 6x+ 10 = 2(x2 + 3x)+ 10 = 2 x+ 3 2 2 − 9 4 + 10 = 2 x + 3 2 2 + 11 . 2 What if there are terms in y as well: x2 + y 2 + 4x − 2y + 3 = x2 + 4x + y 2 − 2y + 3, = (x + 2)2 − 4 + (y − 1)2 − 1 + 3, = (x + 2)2 + (y − 1)2 − 2.

A1 x + a0 where a0 , a1 , . . , an are numbers and an = 0. The degree of a polynomial is simply the highest power of x occurring. Polynomials are probably the most useful functions in mathematics; this is because they are so easy to manipulate. Examples of degree 1 and 3, respectively, are given by f (x) = 5x + 1 and f (x) = 2x3 − 3x + 4. 4 when n = 1, n is even or n is odd. Quadratic functions Quadratic functions are simply polynomials of degree 2. We can use the technique of completing the square to determine the position and shape of the 27 Functions graph: f (x) = = ax2 + bx + c c b a x2 + x + a a a x+ b 2a 2 = a x+ b 2a 2 = − b2 c + 2 4a a − b2 − 4ac 4a2 .