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^ _ g b x ^ _ e d e Z k k Z;Fg T T T a a T F N W F mg kg m s N g g ( ) 0 1078 (11 )( 98 / 2) 107 8 = → = = = = = = T T Fg T (b) a = 0 →T = Fg =107 8 N T Fg T Fg T T (c) a = 0 →T = Fg =107 8 N In all three cases the scale is not accelerating, which means that the two cords exert forces of equal magnitude on it The scale reads the magnitude of either of theseNote that (f B g)(x) ≠ (g B f)(x) This means that, unlike multiplication or addition, composition of functions is not a commutative operation The following example will demonstrate how to evaluate a composition for a given value Example 6 Find (f B g)(3) and (g B f)(3) if f ( x ) = x 2 and g ( x ) = 4 – x2 Solution Step 1 Find (f

F (g (2)), g (x)=2x1, f (x)=x^2 \square!The 3840 was well regarded as an "all around" caliber and was chambered in both rifles and revolvers Whit Collins used 180gn 40″ bullets for the 3840 in his development of the 40 G&A and managed to drive them at just over 1000fps The prototype pistol was created jointly by Irv Stone of BarSto and master gunsmith John FrenchA such that g f = idA Solution For each b 2 B such that b = f(a) for some a 2 A, we set g(b) = a This is wellde ned since for each b 2 B there is at most one such a Now pick some element 2 A and for each b 2

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X!ag(x) = 0 and lim x!af(x) = b, where bis a nite number with b6= 0, Then the values of the quotient f(x) g(x) can be made arbitrarily large in absolute value as x!aand thus 1 the limit does not exist If the values of f(x) g(x) are positive as x!ain the above situation, then lim x!a f(x) g(x)F (x) ∈ F x such that the Galois group of f (x) is isomorphic to G Solution Any group G is a subgroup of a permutation group Sn There exists a field F and a polynomial f (x) (for example general polynomial) with Galois group Sn Let E be the splitting field of f (x) and B = EG Then G is the Galois group of F (x) over B Problem set 11 1B h ʐM R X v ƃ { b g ^ A C e ̐ x ƍ i M ^ x T N b h ʐM @ @ @ @ @ @ @ A } n ̃R X v 葱 _ N i C g ̔M ̋L ^ ł B @ Z ȂǑ ` n R X v A C e ̎ ̎x s Ă ܂

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A,b, then Z b a f(x)dx = F(b)−F(a) where F is any antiderivative of f on a,b Z 3 1 2xdx = x2 3 = 32 −12 = 8 The Second Fundamental Theorem of Calculus Let f be continuous on the closed interval a,b, and define G(x) = Z x a f(t)dt where a ≤ x ≤ b Then G0(x) = d dx "Z x a f(t)dt # = f(x) G(x) = Z x 0 sin2(t)dt G0(x) = sin2(x) H22 3 Continuous Functions If c ∈ A is an accumulation point of A, then continuity of f at c is equivalent to the condition that lim x!c f(x) = f(c), meaning that the limit of f as x → c exists and is equal to the value of f at c Example 33 If f (a,b) → R is defined on an open interval, then f is continuous on (a,b) if and only iflim x!c f(x) = f(c) for every a < c < bH j Z g b a Z p b y g Z Z a _ d e Z k k Z k _ f _ c g u o i j Z a ^ g b d h \, d h g d m j k h \, k

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(b) What are the G x ∑Fy=F N −Mg =0 System #2 topA polyatomic ion is composed of multiple covalently bonded atoms CO₃²⁻ is a polyatomic ion composed of a carbon atom and three oxygen atoms Predict the chemical formula for the ionic compound formed by Au³⁺ and HSO₃⁻ Au (HSO3)3 Predict the chemical formula for the ionic compound formed by NH₄⁺ and PO₄³⁻ (NH4)3PO4B controls G while C controls G and H 4 A controls F and G;

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3For any two numbers aand bwith a@ R ڂ̃o b N w b h A g ^ j Z ^ ̍ @ X n E X 鍂 Z X Ɉ͂܂ꂽ A g ^ ̃i C g C t ̒ S n ł B ׂ̃ m b N X ɂ͍ŏ㋉ ̃V b s O Z ^ m b N X X N G A t B b v X v U ܂ B09 N U E X g c E I u E o b N w b h Ƃ ăG X I X J f ^ A N X ^ ̃o J Ȃǂ o X v ł A I v 11 N ɉ ܂ B The composition of f and g is the function g ∘ f A → C defined by (g ∘ f)(x) = g(f(x)) for all x ∈ A We often refer to the function g ∘ f as a composite function It is helpful to think of composite function g ∘ f as " f followed by g " We then refer

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Get stepbystep solutions from expert tutors as fast as 1530 minutes Your first 5 questions are on us!M Cg down ~a downMake x downward xeq m Cg − T2 = m Ca ⇒ T2 = m C(g − a) = (10 kg)(98 m/s2 − 6125 m/s2) = 367N(Note that while you are holding theF(x)=2x3,\g(x)=x^25,\(f\circ \g)(2) functioncompositioncalculator en Related Symbolab blog posts Intermediate Math Solutions – Functions Calculator, Function Composition Function composition is when you apply one function to the results of another function When referring to

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B controls G and H with H controlled by C Question 3 Develop a network based on the followingDerivative examples Example #1 f (x) = x 3 5x 2 x8 f ' (x) = 3x 2 2⋅5x10 = 3x 2 10x1 Example #2 f (x) = sin(3x 2) When applying the chain rule f ' (x) = cos(3x 2) ⋅ 3x 2' = cos(3x 2) ⋅ 6x Second derivative test When the first derivative of a function is zero at point x 0 f '(x 0) = 0 Then the second derivative at point x 0, f''(x 0), can indicate the type of that pointB and C control G with H depending upon C 5 F and G are controlled by A, G and H are controlled by B with H controlled by B and C 6 A controls F, G and H;

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(2)Suppose that g(x) is a continuous function on an interval a;b such that g(x) >0 for all x Show that Z b a g(x)dx>0 Solution Since g(x) 6= 0 on a;b the function 1 g is de ned and continuous on a;b Hence there is M > 0 so that 1 g(x) < M for all x This means that Definition I21 Let G and H be semigroups A function f G → H is a homomorphism if f(ab) = f(a)f(b) for all a,b ∈ G A one to one (injective) homomorphism is a monomorphism An onto (surjective) homomorphism is an epimorphism A one to one and onto (bijective) homomorphism is an isomorphismOne possible answer is f(n) = 2 b n 1 2 c So now f(1) = f(2) = 2 f(3) = f(4) = 4 f(5) = f(6) = 6 and so on (d)Bijective f(n) = n is boring, but works nicely!

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A M n j b g f ^ y W b @ m K _ M ̖ A N V Y ̋ V ̍U T C g ł BEXAMPLE 43 A rigid beam ABrests on the two short posts shown in Fig 4–8a AC is made of steel and has a diameter of mm,and BD is made of aluminum and has a diameter of 40 mm Determine the displacement of point F on AB if a vertical load ofP yare quadratic surds and if a p x= b p ythen a= band x= y 23 If a;m;nare positivereal numbersanda6=1,thenlog a mn=log a mlog a n 24 If a;m;nare positive real numbers, a6=1,thenlog a m n =log a m−log a n 25 If aand mare

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Then f g(b) = f(g(b)) = f(a) = b, ie f g = idB 119 Show that for an injective function f A !B F(A,B,C,D) = D (A' C') 6 a Since the universal gates {AND, OR, NOT can be constructed from the NAND gate, it is universal(x)g(x)f(x)g!(x) Note For more details about the proof, please consult page 91 of ou rtextbook Example 7 Differentiate with respect to x the function y =(2x1)(x2 2) Example 8 Suppose h(x)=x2 3x2, g(3) = 8, g!(3) = −2, and F(x)=g(x)h(x) Find dF dx # # # # # x=3 42

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E m g b k a l g e k j f z x e d b z z c b z ` ^ a ` z z _ ^ \ n z z y x x w q v u t s p r q q p n o n " , 0 0 3 & / !Tipd4540mm ｽf ｽ ｽ ｽA ｽ} ｽ ｽi ｽC ｽg ｽiAmazonite ｽj ｽ ｽ ｽi ｽ ｽ ｽs ｽ ｽf ｽB ｽX ｽN ( ｽh ｽ ｽi ｽc ｽ^) ｽT ｽC ｽY ｽ ｽ40mm ｽ ｽ ｽl(5)Find counterexamples to each of these statements for f R !R and g R !R (a)If f and g are surjective, then f g is surjective Suppose f(x) = x and g(x) = x Then f g(x) = x

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 NBC, you look like a big company behind a big talent show But have you ever thought about your security on your voting endpoints?V J S E Z y W w Z ރg RCOM x ł̓V J S ̐ Z Љ Ă ܂ B ǂ ́H Ǝv ͑ Ă ܂ B BCBG A Tahari A Parallel Ȃǂ̃f U C i B ܂ A v _ A O b ` A t F f B Ȃǂ̏ i u Ă 邱 Ƃ BA speed of 25 0 m/s and at an angle of 40 degreesa speed of 250 m/s and at an angle of 40 degrees The wall is a distance d=2 m from the release point of the ball (a) How far above the release point does the ball hit the wall?

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Not Mutually Exclusive because P(F ∩ B) = 0077 ≠ 0 Independent because P(FB) = P(F) = 064 9 The following table shows the breakdown of sex and degree among a university's faculty What is the probability that a randomly selected professor is a Male and has a Doctorate P(M ∩ D) = 28 66 b Male or has a Doctorate P(M UG q P G K F G X u y ^ ` j j g i f h d g f d e d OK Y X L z S F l M S H m K N Q J P S G J M J m N M L K J I J v u a J X v G J v X b t ` s j { h f e g h f d e d ProHealth charges $17/per TB test and $40 for Flu vaccine ¡ ¢So we have some tables here that give us what the function f with the functions F and G are when you give it certain inputs so when you input negative for F of negative 4 is 29 that's going to be the output of that function and so we have that for both F and G and what I want to do is evaluate two composite functions I want to evaluate F of G of zero and I want to evaluate G of F of zero so

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