1. Let A= [x ∈ r: x ≠ 1/2] and B= [y ∈ r: ≠ -(1/2)] where r is the set of real numbers. given that f: A → B is defined by f(x) = x/1-2x determine if f(x) is bijective or not.

Q: prove that if (xn) → 2, then (a) (2xn−1/3) → 1; (b) (1/xn) → 1/2. (For this exercise the Algebraic…

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Q: If u=f(x2+2yz,y2+2zx).Prove that

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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…

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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

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A: Given v=lnx-22+y-N2 where N=44. So, v=lnx-22+y-442                                    ........1 To…

Q: Two functions u and v of x and y are said to satisfy the Cauchy- Riemann eguations if u, = v, and u,…

A: Thje given functions are,u(x,y)=y(x2+y2) , v(x,y)=x(x2+y2)

Q: (a) Prove the following relation: B(x + 1, y): B(x,y) x + y %3D

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Q: 3. Show that every y E lị defines a continuous linear functional on l through the inner product,…

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Q: Let F(x, y,z)=xi + yj+zk, f(x,y,z)=||F(x,y,z)||. Prove that: Vf" = nf"-²F F v(In f) =

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Q: -1/2 8. Ž Pn (x) q^ generobr x,2) = (1-2xz + 2?) of legendre polynañials ,oblin the (2n+) xPa W=h+D…

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Q: Prove that f : R – {3} → R – {1}, defined by f(x) = is a bijection. x-1 x-3

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Q: 2. Use Wronskian to show that fi = x, f2 = ln x, and f3 C° (R). x In x are linearly independent in…

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Q: Prove that C[x, y]/ is an integral domain

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Q: Prove or disprove that 0 ≤ x2 - xy + y2 for all x and y in Z.

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A: Given function is: f(x,y,z) = ln(3xy-6) We need to find the domain  for this function.This can be…

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Q: Where x = (X1, X2) E R². 3- N: R2 - → R defined by N(x) = /x1² + x2² is a norm on R². (x1, x2) E R².…

A: Let X=(x1,X2)€R^2

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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: if x, y>o such that x Sy then Vã svy Prove by direct method

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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: If x > 0, prove that T - In (1+ x) > 2+2x

A: Using derivative

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: Which of the following is indeterminate at x = 1? x² + 1 x2 – 1 x2 – 1 Vx +3 - 2 x2 +1 Vx + 3 – 2 x…

A: We plug x=1 in each expression

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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: Let a, y, z be positive real numbers such that r +y+z xyz. Prove that 1 1 1 1 1 1 1+ x2 1+ y2 1+ 22…

A: Given- x + y + z = xyz This inequality can be proved by magic no. Concept-  First we will prove…

Q: If the Wronskian of f and g is t cos t - sin t, and if u = f+ 2g, v = f - g, find the Wronskian of u…

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Q: Prove that for every y E R, there exists a unique x E R such that 2(x – 1)3 – y = 0. %3D

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Q: The domain of the function f (x, y, z) = In(6 – 3ry) is - o {(x, y) E R² : xy > 2} O {(x, y, z) E…

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Q: -x+y<y. 2 If x < y, prove that x <

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Q: Prove that (a) IP{x) = (21 – 1)xP,-1(x) – (I – 1)P;- 2(x), (b) xP(x) – P; -1(x) = IP{{x), (c) P{(x)…

A: Disclaimer:- Since you have posted a question with multiple sub-parts, we will solve first three sub…

Q: Let f(x) = 2x + 1 in Z4 [x]. Then f(x) has a zero Z4 as a subring.

A: Let f(x) =2x+1 in Z4x. Then f(x) has a zero in any ring containing Z4 is a subring_____?____

Q: Show that if AT Ay = y for all y in Rn, then AT A = In.

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Q: Let n E Z. Show that f(x) = x3 – nx +2 is irreducible over Q for %3D n + -1,3, 5.

A: We shall find all n ∈ Z such that f(x) is reducible. Assume that f(x) is reducible. As f(x) is…

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Q: 2. Use Wronskian to show that fi = x, f2 = ln x, and f3 C (R). x In x are linearly independent in

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