Let g : R × N → R be defined by g(x, y) = 3|x|y – 1. Find the image of R x N under g, that is, g(R × N) . Include a proof that your answer is correct.

Q: Let z=x+iy ∈ C where x∈R\{1}, y∈R, and |z|=1. If w = (z+1)/(z-1), then show that Im(w) = y/(x-1).

A: Use the following formulae, to obtain the required result. Real part of a+ib=aImaginary part of…

Q: If the Wronskian of f and g is t cos t - sin t, and if u = f + 3g, v= fg, find the Wronskian of u…

A: It is provided that the Wronskian of f and g is tcost-sint. We need to determine the Wronskian of u,…

Q: Let fn E D[0, 1]. Suppose that fh g on [0, 1], and lim fn(1/2) exists. For each x E [0, 1], show…

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Q: Let A={x e R|-1sx s1} = B, Show that f:A →B given by f (x)=x|x| is a bijecti on.

A: we have to show that given function is bijective

Q: Consider the Cauchy Problem y 0 = a(x) arctan y, y(0) = 1, where a(x) is a continuous function…

A: We have the given Cauchy problem is y'=a(x)tan-1y,  y(0)=1   …..(1) where a(x) is a continuous…

Q: (e) If f(x) is reducible over Q then f(x +a) : a E Z is reducible over Q.

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Q: Prove that the multiplication map · : ℤ/n × ℤ/n → ℤ/n given by [x] · [y] = [xy] is well-defined.

A: A multiplication operation defined on ℤ/nℤ×ℤ/nℤ→ℤ/nℤ as                x·y=xy

Q: Let f(x) = 3sin x for 0 < x S 2 Compute Lf (P) and Uf (P) for the function f and the partition P =…

A: Solve the following

Q: The domain of the function f (x, y, z) = In(6 - 3xy) %3D is {(x, y) E R? : xy 2} {(x, y, z) E R :…

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Q: The domain of the function f (x, y, z) = In(6 – 2æy) is O {(z, y) E R² : zy > 3} O {(z, y, 2) E R° :…

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Q: d x"J,(x)=-x"Jns(x) d, n+1 X.

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Q: Let f(x, y, z) = x³e" Vz² + 5 and let x = rt², y = r2t², and z = sin(t) cos(r). (a) Draw a tree…

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Q: Please solve this question.

A: We are given that L(p(x)) = p(0). Let p(x), q(x) in P4(F) and α, β in the field F. Now,…

Q: Prove that if a e R, then a 0 = 0 = 0 · a. Suppose x, y, z E R. Show that xz = yz if and only if…

A: We will prove the given statements.

Q: 3. Let C[0, 1] be equipped with the inner product (f. g) = f(x)g(x)dx, where f = f(x) and g = g(x).…

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Q: Define f : X → K by f(x) = x'(1), x € X. Again, the linear functional f is not continuous. By 6.4,…

A: Given: f:X→K by fx=x'1, x∈X To prove f is not continuous using the given corollary.

Q: K=Zp(t) ,then f(x)=x^p-t is irreducible over K[x]

A: GIVEN: K = Zp(t)

Q: Consider the function h : Z xZ → Z× Z defined by h(x, у) 3D (2ху, 7x — Зу). (a) Is h injective? (b)…

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Q: Let X be any set and let Y = "2¬X = {x : X → {0, 1}}, that is, Y is the set of all functions from X…

A: To prove that F is a bijective mapping,we have to show that it is both one-to-one and onto.

Q: Let x, y E Rn be given. (a) If ||x|| = ||y|| = 1 and x y = 1, show that x = y. (b) If ||x|| ≤ 1 and…

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Q: Let f,(x) = x E R. Prove that f 0 uniformly on R. COs nr %3D

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Q: Let f : [0, 1]→R be defined by f(x) :=x3 and let P :={0, 0.1, 0.4, 1}. Compute L(P, f) and U(P, f).

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Q: fi (x) = x, f2(x) = cos x, fa(x) = 5

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Q: Let n > 2 be a natural number. Prove that there is no continuous function q : C → C such that…

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Q: Let x, y € R" be given. (a) If ||z||||y||=1 and z-y = 1, show that z = y. (b) If ||z|| ≤ 1 and ||y||…

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Q: ; Let x, y ER" be given. (a) lf ||x|| = ||y|| = 1 and x - y = 1, show that x = y. (b) If ||x|| ≤ 1…

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Q: The function f(x) = x² + 1 is 2 Bijection from R to R Injection from R to R Onto and Injection from…

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Q: Suppose x e R and let A = 1 1 x 1 x x x x x. (a) Compute det A as a function of x. How does the rank…

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Q: Let a function g be defined on R2 with g(x,y) = (xy(x2-y2)exy) / (x2+y2) Use the following to…

A: Limiting value of g.

Q: use the inner product = p(x)q(x)dx. to find the projection of p(x) q(x) = 5x + 2. = S'; x3 onto

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Q: 2. Let f : R × N → R be defined by f(x, y) = 2|x|y+ 1. Find the image of R×N under f, that is,…

A: This is a problem of set and function.

Q: Let f : (0, 1] → (0,1) be the function defined by f ) = „ for all n E N, and f(x) = x if æ E (0, 1]…

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Q: Define T E L(C') by T(z1, z2, Z3, Z4, Z5, Z6, Z7) = (z3, Z4, Z5, Z6, Z7, 0, 0). (a) Prove that there…

A: Let the standard basis for C7 (complex basis) is given as:

Q: Prove that H(x, y) = (x, –y) provides a conjugacy between 1 X' = 1 X and Y' Y. 1 1 1 1

A: We are Given that :  H(x,y)=(x,-y) X'=11-11X , Y'=1-111Y The Matrix H is corresponding to the…

Q: The mapping w = z² + 1 is not conf ormal at O z = 0 O z = i O z = -i

A: We know that analytical function is not conformal at critical points. Let's find the critical points…

Q: For x, y, z, w > 0, prove that (x/z)+(y/z)+(z/w)+(w/x)≥4

A: It should by x/y, instead of x/z

Q: Let C[0, 1] have inner product [f(x), g(x)] = ∫ f(x)g(x) dx. Find α so that (x + αx2 ) ⊥ (x + 1)

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Q: Find det (T) for T:R- R, where T(x,,×2,*3) = (x,- x2, X2 - X3, X3-x).

A: We will find the matrix [T] from the image of the standard basis vectors of R³ under the…

Q: For x = 1, 2, 3 and y = 1, 2, let the joint pmf of X and Y be defined by fX,Y (x, y) = x + y/21…

A: Solution

Q: Let f(x) = sin x and g(x) = cos x, and W = span{f,g}. Use the Wronskian to show that f and g are…

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Q: Let g : Zx Z-ZxZ be the function given by g(x,y) = (3x + Y, x – y). Find, with proof, an element of…

A: Take a point (1, 0) and suppose it belongs to the range set.

Q: The domain of the function f (x, y, z) = In(6 – 3xy) %3D is {(x, y) E R² : xy 2} {(x, y, 2) E R³ :…

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Q: 41. Let f(x, y, z) =x?y+ y²z. Use the Chain Rule to calculate af/as and af/at (in terms of s and f),…

A: Since f is function of x, y and z.  For partial derivative f with respect to s and t we using chain…

Q: Let f:C → C be an entire function. Show that the function g:C \ {0} → C defined by g(z) = f(1/7) is…

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Q: Let f(x, y, z) = x2y + y2z. Use the Chain Rule to calculate af /as and af /at (in terms of sand t),…

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Q: Define g : Z → Z × Z by if t is even if t is odd S (t, t + 3) g(t) = I (t, 2) Is g injective? Is g…

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