Theorem 4 Theorem 5 4 Theorem If f and y are continuous at a and e is a constant, then the following 9 Theorem (a) Any polynomial h ontinuus erywhee the is, continus on R-- h) Any rational function is cotimuces wherever it is defned that is, it is conin- unas an its domain funetions are also continuous at a 21- A ef PROOF (a) A polyaomial is a function of the form PROOF Each of the five parts of this theorem follows from the corresponding Limit Law in Section 2.3. For instance, we give the proof of part 1. Since f and g are contir ous at a, we have where are constants. We know that lim by Law 7) lim f(x) - fla) and lim g(x) – gla) m- 1,2. .. and (by ) This equation is precisely the statement that the function f(x) -is a continuous function. Thus, by part 3 of Theorem 4, the function g(x) - cr" is continuous. Since P is a sum of functions of this form and a constant function, it follows from part I of Therefore Theorem 4 that P is continuous. lim (f+ 9)(x) = lim [Sx) + g(x)] (b) A rational function is a function of the form lim f(x) + lim g(x) (by Law 1) Pa) S - where Pand Q are polynomials. The domain of f is D= {xER| Q) - 0). We know from part (a) that Pand Q are continuous everywhere. Thus, by part 5 of Theo- rem 4, f is continuous at every number in D. - Se) + g(a) =(S+ 9(a) As an illustration of Theorem 5, observe that the volume of a sphere varies con- timuously with its radius because the formula Vir) -jer' shows that V is a polyno- mial function of r. Likewise, if a ball is thrown vertically into the air with a velocity of 50 /s, then the height of the ball in feet i seconds later is given by the formula h- S0 - 16r. Again this is a polynomial fanction, so the height is a continuous fune- tion of the elapsed time, as we migh expect. This shows that /+ g is contimucus at a It follows from Theorem 4 and Definition 3 that if f and g are contimmous on an inter- val, then so are the functions f+ g.f-a.cf. fo, and Gif g is never 0) flg. The following theorem was stated in Scction 2.3 as the Direet Substitution Property. Theorem 9 9 Theorem If g is continuous at a and f is continuous at g(a), then the com- posite function fog given by (f. g)(x) - f(g(x)) is continuous at a. This thcorem is often expressed informally by saying "a continuous function of a continuous function is a continuous function." Theorem 7 PROOF Since g is continuous at a, we have lim g(x) = g(a) The inverse trigonometric fanctions are reviewed in Section 1.5. 7 Theorem The following types of functions are continuons at every number in their domains: Since f is continuous at b = g(a), we can apply Theorem 8 to obtain • polynomials • trigonometric funetions • exponential functions • root functions • inverse trigonometric funetions • logarithmic functions • rational functions lim f(g(x)) – f(g(a)) which is precisely the statement that the function h(x) - f(g(x)) is continuous at a; that is, fog is continuous at a. M(x) = 1+

Theorem 4 Theorem 5 4 Theorem If f and y are continuous at a and e is a constant, then the following 9 Theorem (a) Any polynomial h ontinuus erywhee the is, continus on R-- h) Any rational function is cotimuces wherever it is defned that is, it is conin- unas an its domain funetions are also continuous at a 21- A ef PROOF (a) A polyaomial is a function of the form PROOF Each of the five parts of this theorem follows from the corresponding Limit Law in Section 2.3. For instance, we give the proof of part 1. Since f and g are contir ous at a, we have where are constants. We know that lim by Law 7) lim f(x) - fla) and lim g(x) – gla) m- 1,2. .. and (by ) This equation is precisely the statement that the function f(x) -is a continuous function. Thus, by part 3 of Theorem 4, the function g(x) - cr" is continuous. Since P is a sum of functions of this form and a constant function, it follows from part I of Therefore Theorem 4 that P is continuous. lim (f+ 9)(x) = lim [Sx) + g(x)] (b) A rational function is a function of the form lim f(x) + lim g(x) (by Law 1) Pa) S - where Pand Q are polynomials. The domain of f is D= {xER| Q) - 0). We know from part (a) that Pand Q are continuous everywhere. Thus, by part 5 of Theo- rem 4, f is continuous at every number in D. - Se) + g(a) =(S+ 9(a) As an illustration of Theorem 5, observe that the volume of a sphere varies con- timuously with its radius because the formula Vir) -jer' shows that V is a polyno- mial function of r. Likewise, if a ball is thrown vertically into the air with a velocity of 50 /s, then the height of the ball in feet i seconds later is given by the formula h- S0 - 16r. Again this is a polynomial fanction, so the height is a continuous fune- tion of the elapsed time, as we migh expect. This shows that /+ g is contimucus at a It follows from Theorem 4 and Definition 3 that if f and g are contimmous on an inter- val, then so are the functions f+ g.f-a.cf. fo, and Gif g is never 0) flg. The following theorem was stated in Scction 2.3 as the Direet Substitution Property. Theorem 9 9 Theorem If g is continuous at a and f is continuous at g(a), then the com- posite function fog given by (f. g)(x) - f(g(x)) is continuous at a. This thcorem is often expressed informally by saying "a continuous function of a continuous function is a continuous function." Theorem 7 PROOF Since g is continuous at a, we have lim g(x) = g(a) The inverse trigonometric fanctions are reviewed in Section 1.5. 7 Theorem The following types of functions are continuons at every number in their domains: Since f is continuous at b = g(a), we can apply Theorem 8 to obtain • polynomials • trigonometric funetions • exponential functions • root functions • inverse trigonometric funetions • logarithmic functions • rational functions lim f(g(x)) – f(g(a)) which is precisely the statement that the function h(x) - f(g(x)) is continuous at a; that is, fog is continuous at a. M(x) = 1+

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section5.1: Inverse Functions

Problem 18E

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