Various Important Classes of Functions (Elementary Functions)
In this chapter, we will familiarize ourselves with the most commonly occurring functions in mathematics and in applications of mathematics to the sciences. These are the polynomials, rational functions, exponential, power, and logarithm functions, trigon
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Various Important Classes of Functions (Elementary Functions)
In this chapter, we will familiarize ourselves with the most commonly occurring functions in mathematics and in applications of mathematics to the sciences. These are the polynomials, rational functions, exponential, power, and logarithm functions, trigonometric functions, hyperbolic functions, and their inverses. We call the functions that we can get from the above functions using basic operations and composition elementary functions.
11.1 Polynomials and Rational Functions We call the function p : R → R a polynomial function (a polynomial, for short) if there exist real numbers a0 , a1 , . . . , an such that p(x) = an xn + an−1 xn−1 + · · · + a1 x + a0
(11.1)
for all x. Suppose that in the above description, an = 0. If x1 is a root of p (that is, if p(x1 ) = 0), then p(x) = p(x) − p(x1 ) = an (xn − x1n ) + · · · + a1 (x − x1 ). Here using the equality xk − x1k = (x − x1 )(xk−1 + x1 xk−2 + · · · + x1k−2 x + x1k−1 ), and then taking out the common factor x − x1 we get that p(x) = (x − x1 ) · q(x), where q(x) = bn−1 xn−1 + · · · + b1 x + b0 and bn−1 = an = 0. If x2 is a root of q, then by repeating this process with q, we obtain that p(x) = (x − x1 )(x − x2 ) · r(x), where r(x) = cn−2 xn−2 + · · · + c1 x + c0 and cn−2 = an = 0. It is clear that this process ends in at most n steps, and in the last step, we get the following. © Springer New York 2015 M. Laczkovich, V.T. S´os, Real Analysis, Undergraduate Texts in Mathematics, DOI 10.1007/978-1-4939-2766-1 11
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11 Various Important Classes of Functions (Elementary Functions)
Lemma 11.1. Suppose that in (11.1), an = 0. If p has a root, then there exist not necessarily distinct real numbers x1 , . . . , xk and a polynomial p1 such that k ≤ n, the polynomial p1 has no roots, and p(x) = (x − x1 ) · . . . · (x − xk ) · p1 (x)
(11.2)
for all x. It then follows that p can have at most n roots. The above lemma has several important consequences. 1. If a polynomial is not identically zero, then it has only finitely many roots. Clearly, in the expression (11.1), not all coefficients are zero. If am is the nonzero coefficient with largest index, then we can omit the terms with larger indices. Then by the lemma, p can have at most m roots. 2. If two polynomials agree in infinitely many points, then they are equal everywhere. (Apply the previous point to the difference of the two polynomials.) 3. The identically zero function can be expressed as (11.1) only if a0 =. . .=an = 0 (since the identically zero function has infinitely many roots). 4. If in an expression (11.1), an = 0 and the polynomial p defined by (11.1) has an expression of the form p(x) = bk xk + bk−1 xk−1 · · · + b1 x + b0 , where bk = 0, then necessarily k = n and bi = ai for all i = 0, . . . , n. We see this by noting that the difference is the identically zero function, so this statement follows from the previous one. The final corollary means that a not identically zero polynomial has a unique expression of the form (11.1) in wh
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