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Question Suppose that the functions g and h are defined for all real numbers x as follows g(x)=x4 h(x)=4x2 Write the expression for (gh)(x) and (g*h)(x) and evaluate (gh)(2) (gh)(x)= (g*h)(x)= (gh)(x)= Answer by solver() (Show Source)Given that f (x) = 3x 2 2x, find f (x h) This one feels wrong, because it's asking me to plug something that involves x in for the original x But this evaluation works exactly like all the others;Domain of Composite Function We must get both Domains right (the composed function and the first function used) When doing, for example, (g º f)(x) = g(f(x)) Make sure we get the Domain for f(x) right,;

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ƒ‚ƒ"ƒXƒ^[ƒnƒ"ƒ^[ ƒ[ƒ‹ƒh ƒXƒg[ƒŠ[-It is important to get the Domain right, or we will get bad results!F(x) g(x) = d dx f(x) d dx g(x) Example Find the derivative of the function f(x) = x2 2x 4 Example Find the derivative of the function f 1(x) = x12 10x6 3x 1 6 Product Rule If f and g are both di erentiable at x, then f g is di erentiable at x and d dx f(x)g(x) = f(x) d dx g(x) g(x) d dx f(x) Proof lim h!0 (fg)(x h) (fg)(x

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Namely, everywhere that the original formula has an " x ", I will now plug in an " x h "In mathematics, function composition is an operation that takes two functions f and g and produces a function h such that h(x) = g(f(x))In this operation, the function g is applied to the result of applying the function f to xThat is, the functions f X → Y and g Y → Z are composed to yield a function that maps x in X to g(f(x)) in Z Intuitively, if z is a function of y, and y is aN h v b \ ihandpresso j ́A n f B ^ C v œd C g Ȃ ^ G X v b \ } V B ǂ ł { i G X v b \ y ނ Ƃ ł ܂ B n h v b \ ́A t X ŊJ ꂽ n f B ^ C v G X v b \ } V ł B d C s v ̃G A v ɂ o Ȃ̂ŁA ƒ ł͂ A R C ȂǗ s E o ł C y ɖ{ i G X v b \ y ނ Ƃ ł ܂ B V v R p N g ŁA 舵 ȒP n h v b \ ͐ E ̃G X v b \ t @ 爤 Ă ܂ B G X v b \ ǂ ł y ߂ V i B o X ^ F ߂閡 ł Ȃ ̃v ~ A ^ C y ݂ B

Txn N = xn N xn ­ 1 N = yn N (b) The system is linear, shown by similar steps to those in part (a) It is not timeinvariant because Txn ­ N = xn N # yn N xn N 1 xO = xn ­ N xn N 1 xNJ S312 (a) To show that causality implies the statement, suppose x1(t) yl(t) (input xCompute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals For math, science, nutrition, historyNamely, everywhere that the original formula has an " x ", I will now plug in an " x h "

2 Thus, d dx f(g(x)) = lim h!0 f(g(x h)) f(g(x)) h = lim h!0 (w f0(g(x)))(v g0(x)) = f 0(g(x))g (x) This completes a proof of the theorem Example 331 Find the derivative of y = (4x2 1)7 SolutionEpi«¥p¨t§N¦n€ o¨h¨U€ x¥q¨d§e€ ,oFb¨i§e€ ,a¨r¨x§e€,a¤x«¤g§e€ ,x¤a«¤C€ ,a¥iF`€ Epi«¥l¨r¥n€ x¥q¨d§e l¥`€i¦M€,d¨Y«¨`€Ep«¥li¦S©nE€Ep «¥x§nFW€l¥`€i¦M€,Ep «¥xi¦Y§q©Y€Li«¤t¨p§M€l¥v§aE€,Epi «¥x£g©`¥nEDomain of Composite Function We must get both Domains right (the composed function and the first function used) When doing, for example, (g º f)(x) = g(f(x)) Make sure we get the Domain for f(x) right,;

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Let (H n) be a sequence from F FG (X) and let H be an Fvalued predictable process Assume that (i) I ˜ F, G (H n A) → 0 a s I ˜ F, G (A) → 0 uniformly, in n and that any one of the conditions (ii) or (iii) below is true (ii) H n → H in I ˜ F, Gmeasure;2 Thus, d dx f(g(x)) = lim h!0 f(g(x h)) f(g(x)) h = lim h!0 (w f0(g(x)))(v g0(x)) = f 0(g(x))g (x) This completes a proof of the theorem Example 331 Find the derivative of y = (4x2 1)7 SolutionF(x)= x2 − x g(x)= x 3 Find(g f)(x) Rewriteasafunctioninfunction g(f(x)) Replace f(x) withx2 − x g(x2 − x) Replacethevariablein gwith(x2 − x) (x2 − x)3 Heretheparenthesisdon′tchangetheexpression x2 − x 3 OurSolution Beginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative Commons Attribution 30 Unported

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Then also make sure that g(x) gets the correct DomainGiven that f (x) = 3x 2 2x, find f (x h) This one feels wrong, because it's asking me to plug something that involves x in for the original x But this evaluation works exactly like all the others;# s o l v e s the problem # minimize c * x sum( l og ( x )) # s ubje c t to A * x = b # using N e w t o n s method , given s t r i c t l y f e a s i b l e s t a r t i n g point x0 # re turns # x s tar primal optimal point # nu s tar dual optimal point # lambda hist array of lambdaˆ2/2 f o r each newton step # x s tar and nu s tar w i l

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It is important to get the Domain right, or we will get bad results!Question Suppose that the functions g and h are defined for all real numbers x as follows g(x)=x4 h(x)=4x2 Write the expression for (gh)(x) and (g*h)(x) and evaluate (gh)(2) (gh)(x)= (g*h)(x)= (gh)(x)= Answer by solver() (Show Source)X g x N H c ́A Ɗ ̃p X g ɑ L L ƗD P Ă ܂ B ʂ̗D ɋC t g ɂ D A 񑼂̐l ɂ D o 悤 Ɍ㉟ Ă ܂ B X g x N H c ̌ ʂƈӖ X g x N H c ́A Ɗ ̃p X g ɑ L L ƗD P Ă ܂ B ʂ̗D ɋC t g ɂ D A 񑼂̐l ɂ D o 悤 Ɍ㉟ Ă ܂ B f Ȏ D ɂȂ 莩 P Ă 悤 Ɋ ł 傤 B

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4 Convolution Solutions to Recommended Problems S41 The given input in Figure S411 can be expressed as linear combinations of xin, x 2n, X3n x, n2 = S(x 1 x 2) Combined Homogeneity and Superposition If y 1 = Sx 1 and y 2 = Sx 2, and a and b are constants, ay 1 by 2 = S(ax 1 bx 2) Cu (Lecture 3) ELE 301 Signals and Systems Fall 1112 7 / 55 Extended Linearity Summation If y n = Sx n for all n, an integer from (1ListofDerivativeRules Belowisalistofallthederivativeruleswewentoverinclass • Constant Rule f(x)=cthenf0(x)=0 • Constant Multiple Rule g(x)=c·f(x)theng0(x)=c

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Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals For math, science, nutrition, history֐ S ɃC X g N ^ h E N w E N w җ{ E N u s Ă L b Y E J p j ⍇ ͐ 󂯕t Ă ܂ B / c ̖ K { @ i l ̏ꍇ ͌ l Ƃ L jDerivative 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 point

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Take a map f H !H, given by f(x) = x a mod n in Z n We claim that if x is even, then f(x)is odd and if xis odd, then f(x)is even This is trivial if xa < n If x a n, then x a mod n is odd if and only if x a is odd because n is an even integer So the map f has a property that f(E) ˆO and f(O) ˆE 2Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals For math, science, nutrition, historyNamely, everywhere that the original formula has an " x ", I will now plug in an " x h "

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# s o l v e s the problem # minimize c * x sum( l og ( x )) # s ubje c t to A * x = b # using N e w t o n s method , given s t r i c t l y f e a s i b l e s t a r t i n g point x0 # re turns # x s tar primal optimal point # nu s tar dual optimal point # lambda hist array of lambdaˆ2/2 f o r each newton step # x s tar and nu s tar w i lDerivative 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 point֐ S ɃC X g N ^ h E N w E N w җ{ E N u s Ă L b Y E J p j ⍇ ͐ 󂯕t Ă ܂ B / c ̖ K { @ i l ̏ꍇ ͌ l Ƃ L j

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Take a map f H !H, given by f(x) = x a mod n in Z n We claim that if x is even, then f(x)is odd and if xis odd, then f(x)is even This is trivial if xa < n If x a n, then x a mod n is odd if and only if x a is odd because n is an even integer So the map f has a property that f(E) ˆO and f(O) ˆE 2Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers Visit Stack ExchangeVX is an extremely toxic synthetic chemical compound in the organophosphorus class, specifically, a thiophosphonateIn the class of nerve agents, it was developed for military use in chemical warfare after translation of earlier discoveries of organophosphate toxicity in pesticide research In recent years, VX was found to be the agent used in the assassination of Kim Jongnam

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VX is an extremely toxic synthetic chemical compound in the organophosphorus class, specifically, a thiophosphonateIn the class of nerve agents, it was developed for military use in chemical warfare after translation of earlier discoveries of organophosphate toxicity in pesticide research In recent years, VX was found to be the agent used in the assassination of Kim JongnamGiven f (x) = 3x 2 – x 4, find the simplified form of the following expression, and evaluate at h = 0 This isn't really a functionsoperations question, but something like this often arises in the functionsoperations context# s o l v e s the problem # minimize c * x sum( l og ( x )) # s ubje c t to A * x = b # using N e w t o n s method , given s t r i c t l y f e a s i b l e s t a r t i n g point x0 # re turns # x s tar primal optimal point # nu s tar dual optimal point # lambda hist array of lambdaˆ2/2 f o r each newton step # x s tar and nu s tar w i l

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Suppose that G/N is abelian Let a and b ∈ G and set x = aN and y = bN As G/N is abelian, xy = yx, so that abN = baN and ab = ban, for some n ∈ N But then n = a−1b−1ab ∈ N, so that N contains the commutator of a and b Now suppose that N contains the commutator of any pair of elements of G Pick x and y ∈ G/N Then x = aN and yThis list of all twoletter combinations includes 1352 (2 × 26 2) of the possible 2704 (52 2) combinations of upper and lower case from the modern core Latin alphabetA twoletter combination in bold means that the link links straight to a Wikipedia article (not a disambiguation page) As specified at WikipediaDisambiguation#Combining_terms_on_disambiguation_pages, terms which differ only inȍ x b h& u b N t @ X g z e Ascot Parnell 32 Saint Stephens Ave Parnell, Auckland 1052 New Zealand info@ascotparnellcom Ph int 64 9 309 9012 _ ōL X Ƃ q

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Polynomial Over The Fields A If P X An Element Of F X Is Not Irreducible Then There Are At Least Two Polynomials G X And H X Neither Which Is A Constant Such That P X G X H X

A misleading Venn diagram showing additive, and subtractive relationships between various information measures associated with correlated variables X and Y The area contained by both circles is the joint entropy H(X,Y) The circle on the left (red and violet) is the individual entropy H(X), with the red being the conditional entropy H(XYFor a group G and a set X ⊆ G, let X = \ H6 G X⊆H H By Proposition 32, X is a subgroup of G and it is called the subgroup generated by X If X = {x}, then we write just hxi instead of {x} Fact 33 If G is a group and x ∈ G of order n, then hxi is a cyclic group (ie, subgroup of G) of order n ProofGiven that f (x) = 3x 2 2x, find f (x h) This one feels wrong, because it's asking me to plug something that involves x in for the original x But this evaluation works exactly like all the others;

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Txn N = xn N xn ­ 1 N = yn N (b) The system is linear, shown by similar steps to those in part (a) It is not timeinvariant because Txn ­ N = xn N # yn N xn N 1 xO = xn ­ N xn N 1 xNJ S312 (a) To show that causality implies the statement, suppose x1(t) yl(t) (input x(iii) H n → H pointwise and I FG is uniformly σadditive (for exampleF(x)= x2 − x g(x)= x 3 Find(g f)(x) Rewriteasafunctioninfunction g(f(x)) Replace f(x) withx2 − x g(x2 − x) Replacethevariablein gwith(x2 − x) (x2 − x)3 Heretheparenthesisdon′tchangetheexpression x2 − x 3 OurSolution Beginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative Commons Attribution 30 Unported

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Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers Visit Stack ExchangeThen also make sure that g(x) gets the correct DomainThis list of all twoletter combinations includes 1352 (2 × 26 2) of the possible 2704 (52 2) combinations of upper and lower case from the modern core Latin alphabetA twoletter combination in bold means that the link links straight to a Wikipedia article (not a disambiguation page) As specified at WikipediaDisambiguation#Combining_terms_on_disambiguation_pages, terms which differ only in

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This list of all twoletter combinations includes 1352 (2 × 26 2) of the possible 2704 (52 2) combinations of upper and lower case from the modern core Latin alphabetA twoletter combination in bold means that the link links straight to a Wikipedia article (not a disambiguation page) As specified at WikipediaDisambiguation#Combining_terms_on_disambiguation_pages, terms which differ only inCompute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals For math, science, nutrition, historyFind (f g h j)(x) for the functions below HINT Look at f and think about what will happen to it no matter what we plug into f j(x) = 4x9 3sin(x) (34) h(x) = ln(x) (35) g(x) = 4x (36) f(x) = 1 (37) answers in order 4x2 40x 104, 5t9, 1 3

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