Let u be a root of ƒ = t³ − t² +t+ 2 € Q[t] and K : = (a) Show that f = mo(u). (b) Express (u²+u+1)(u² Q(u). −u) and (u−1)−¹ in the form au²+bu+c, for some a, b, c € Q.

Q: Let R={(x,y) ERxR: |x| = |v|). Then R a function on R. оо True False

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A: we have to find the domain for f(x)

Q: a) f(AUB) = f(A) U ƒ(B). b) (i) f(An B) ≤ f(A)n ƒ(B). (ii) Find a function f {1,2} → {a}, and A, B C…

A: a)  We  have  to prove  that  Now,      Thus,  (Proved)[since, b) i) We  have  to show  that Let  be…

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A: To solve the following

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Q: Solve ux – uy +yu = 0, where u is a function of x and y. %3D

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Q: Use the function to find (a) the image of v and (b) the preimage of w.

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Q: 4. A) Factor f(x) into linear factors given that k is a zero of f(x). B) Find all roots of the…

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Q: Suppose f: X→Y, and A, B CY. Prove that (a) f-¹(AUB) = ƒ−¹(A) U ƒ−¹(B). (b) f-¹(ANB) = f¯¹(A) n…

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Q: Prove that bx q = b. ((Vt) - Vt), where q = V x t

A: We have to prove b→×q→=b→·∇t→Tr-∇t→ for q→=∇×t→ b→×q→=b→×∇×t→          =b→×∑i^∂∂ti×t→…

Q: b) Let f: D→ C be a function and Y a subset of C. Show that f¹ [C-Y] =D-f¹ [Y].

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Q: How many functions f:R→R  exists such that : 2f(x)+2f(y)=f(x+y)+f(x−y)+2y holds for all real x and…

A: By choosing x=y=0. Also by choosing x=y we can see f(2x)=4f(x).  Moreover choose x=0 and use f(0)=0,…

Q: n For each nEN, consider x = with x ER+. Define the sequence {gn} by x (a) (b) A = [1,2) CR. (1)…

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Q: Let f: Z → Z be defined by f(x) = x² + 1, and let C = {1,2,3}.

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Q: Using the substitution u = (A) (B) C D 1+1 du [₁ -1/2 du [(u ¹/2 - u-1/2) d du (u - 1) / u-²1) x + 1…

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Q: u = [2, 6, –5] and v= [-3,–1,6] %3D A · A + (A + ng-) · (A x n)

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Q: 6. Prove that |xy| = |x|ly| for all real numbers x, y.

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Q: (2) Let f(x) = ax³ + bx? + cr + d, where a, b, c, d e R and a + 0. Prove that f(x) must have a root.…

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Q: 5. Show that the functions x, x² interval [-1,1]. x are mutually orthogonal in the -

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Q: Show that f : ℝ → ℝ, f(x) = 5x − 3 is a bijective function.

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Q: 04 Let f(x) be a contineous function such that Ler - 5/W)dx = = 4 and =9, then a) 81 b) 101

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Q: c) Which of the following functions are analytic in 121² 2 2: 42²26649 (ii) Z²-1 Z-√√Z²74

A: