(a) Let f, 9, h : N → R²º. Prove or disprove that if f + g€ O(h), then f e O(h) and gE O(h). (b) For all functions f, g : N→ R2º, define the product function f · g : N → R2° as follows: Vn E N, (f - g)(n) = f(n) · g(n). Let f, g,h : N → R²º. Prove or disprove that if f ·g€ O(h), then ƒ € O(h) and E O(h).

Q: Let f : R R:f(x) = 10x + 7. Find the function g : R→R such that gof = fog = IR. %3D %3D

A: Given f: R -> R  such that f(x) = 10x + 7 and  g : R -> R such that g o f = f o g = IR

Q: f: R2 → R, f(x1, x2) = x1-x2 function is given. a) Can f be an surjective function? b) Can f be an…

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Q: Suppose that f: A → B is a function, C ≤ B and D C B. Prove that ƒ¯¹(C − D) = ƒ¯¹(C) − ƒ¯¹(D) -

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A: Given:

Q: 1. Let f : Z6 → Z6 be such that f(7) = x2 + 3. (a) Is f a well defined function? (b) Is f…

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Q: Let f e E(0, ox) and let a be a positive real number. Define the function fa as follows Sal) = {";…

A: Properties of Laplace transformation

Q: (b) Define a function F:F: R XR → R XR as follows: For all (x, y) ERXR, (F(x, y) = (2x + y, 2x - y)…

A:

Q: 5. Consider the function f : Z × Z → Z × Z defined as f(n, m) = (n + m, 3n + 4m). (a) Find (n, m) E…

A: As per our company guidelines we are supposed to answer only first 3 sub-parts. Kindly repost other…

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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Q: Let f : Z+ → N be the floor function f(x) = [(x – 1)/2].

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Q: Which of the following is the inverse of the functionf : [0, ∞) → [3, 0), f(x) = 2.x² + 3? Select…

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Q: Q2\A\ Which of the following functions are bijective 2- f: R{0}→ R\{0},f(x) = 1/x این 2- f:R→R,f(x)…

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Q: . Consider the function f: N – N defined by f(n) = n² . Prove that f has no right inverse, and…

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Q: 2. ( images. (a) f({0, 1, 2}) Let f: Z→ R, with f(n) = 4-n². Find the following direct and inverse…

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Q: Show that the following relations are not functions on R. a) ƒ = {(x, y) = RxR | y=x/(x² −7x+12)}

A: Given function. y=xx2-7x+12 is not define for x2-7x+12=0 =&gt; (x-3)(x-4)=0

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: 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: Show that for any function f : X → Y , there are two functions g : S → T and h : X → S such that (a)…

A: To Show: Show that for any function f : X → Y, there are two functions g: S → T and h: X → S such…

Q: Let f : R → R be a function given by f(x) = 2x – 1 (a) Determine whether f is bijective or not. (b)…

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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: Suppose f: X→ Y, and A, B CY. Prove that (a) f¹(AUB) = f¹(A) U f¹(B).

A: Union of two sets A and B is denoted as A∪B  It contains all the elements that are either in A or in…

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

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Q: f: R2 → R, f(x1, x2) = x1-x2 function is given. a) Can f be an surjective function? b) Can f be an…

A: f: R2 → R, f(x1, x2) = x1-x2 function is given. a) Can f be an surjective function? b) Can f be an…

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Q: Prove or disprove: f : Z30 → Z15 given by f([x]) = [x²] is a well-defined function.

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Q: Iff: X→Y is an injective function and A, BCX then: f(A-B) f(A)- f(B)

A: Use the concept of injection map of functions.

Q: Exercise 2. Let f : A → B be a function with Y1 and Y2 subsets of B. Prove that if Y1 C Y2, then…

A: Let f:A→B be a function and let Y1,Y2⊆B. Suppose that Y1⊆Y2 To show f-1(Y1)⊆f-1(Y2) Let x∈f-1(Y1)…

Q: f: R → R f(x) = -5x Select the correct description of the function f. O Neither one-to-one nor onto…

A: Given function is f: ℝ → ℝf(x) = -5x

Q: Show that the function f: N² → Z given by f(m, n) = m²-n² is neither injective: suriective. nor

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