) Let f: A-B be a function which is not surjective and g : B C which is bijective. Prove that g of: A C is not surjective.

Q: Prove the following Quotient Rule, where f, g are differentiable:

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Q: 4. If f:(X,r)(Y.A) and g:(Y.A) (Z,r)are closed functions, prove that g of :(X.r)(Z.r) is closed…

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Q: Prove that the function f : R → R, given by f(x) = 2x, is one-one and onto.

A: Given, function f : R → R, given by f(x) = 2x.

Q: Determine if the function f : RxR → R, defined by f(x, y) = x2", for all (x, y) E R×R, is…

A: f is not one-one  But, f is onto.

Q: Show that, for any set A, there is exactly one map f from the empty set ∅ to A. When is f injective?…

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Q: Find a codomain B such that ƒ: A → B, ƒ(x) = e²´ is surjective. Show that g: B → A, g(x) = Vln x is…

A: As you asked only subpart (C) and (D) So, I answered here only subpart (C) and (D).

Q: let P: XY be a function prove that Phas an inverse if and only if fis abijection.

A: Let us define a bijective functions. We know that , a function f : X → Y is bijective if it is both…

Q: Determine whether the function f: R- R, f(x) =,x # 3 is surjective. x+1 %3D r-3

A: Consider the provided question, We need to determine whether the function f:R→R fx=x+1x-3, x≠3 is…

Q: Let X,Y,Z be sets and f:X→Y, g:Y→Z. Prove that if g is injective and g◦f is surjective,then f is…

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Q: Let ƒ be a mapping from [2, +∞ [to, +∞[defined by: 3 f(x) = x + = x f has a fixed point. O False O…

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Q: Suppose and z are functions of x and y, then prove.

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Q: Prove that the function g: R > R defined by f(x) = 1+2^x is one-to-one but not onto.

A: Consider the function f:R->R defined by:

Q: 3. Let f: A B and g B → A be functions such that gof = i4. Prove that if g is injective, then f is…

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Q: 4. Show whether or not f is bijective. Letf : R → R_be defined as f(x) = I피+1 %3D

A: A function f is said to bijective if it is one-one and onto

Q: Prove that the function f: R –→ R defined by f (x) = 3x – 5 is bijective.

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Q: Exercise 3 Define f : Rª\{0} by f(r) = |r|,'. Using the e-5-definition, prove that f is continuous.…

A: We are given a function f whose domain is ℝd-{0}. The function f is defined as  f=x2-1. Where…

Q: Given a function f: A → B and subsets X,Y C B, prove that f"(XNY) = f(X)nf-(Y).

A: He we will prove the given result.

Q: Let f, g,h: A→ A be any bijective functions. Then (f ogoh)-

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Q: Prove that the function f: R -→ R defined by f(x)= x³ is bijective.

A: If a function is both one-one and onto, then the function is bijective.

Q: If a is transcendental over F, show that every element of F(a) that isnot in F is transcendental…

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Q: 1. Consider the functions f : A → B and g : B → C. (a) If f and g are both surjective, show that so…

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Q: Suppose that f : A → B and g : B → C are both injective functions, prove that the function go f : A…

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Q: Let f.g.h:A - A be any bijective functions. Then (g oh on-

A: (g∘h∘f)-1=f-1∘h-1∘g-1 composition of function is associative…

Q: Let f:Z → Z be defined by f (x) = 2x + 1 where x € Z. Is f a one to one correspondence (bijection)?

A: Now let

Q: Let f:R-R defined by f(x) =x² , then f ( x) is bijective صواب ihi

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

A: We have give Beta function relation, we have to show left hand side is equal to right hand side.

Q: Prove or provide a connter C are functions such that gois one-to-one. then f is one-to-one. xample…

A: See the detailed solution below.

Q: Let f : A → B and g : B C be surjective functions. Prove or disprove whether gof is surjective.

A: Here given that                       f :A→B   and    g :B→C  be both surjective function .  Now we…

Q: Consider a function f:C C, defined by f(z) = 2z2 Then which of the following is true? Check all that…

A: Injective ( one to one): If x1 not equal to x2 then fx1 not equal to fx2. If x1 = x2 then fx1 = fx2.…

Q: a. The function f : Z → Z, f(x) = 2x is onto. (A) True (В) False b. The function f : Z → Z, f(x) =…

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Q: Let f : R → R be continuous such that f(q) = 0 for all q ∈ Q. Prove that f must be the zero function

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Q: 7. If a functionf defined in [a, b] is such that o (i) f is derivable in [a, b], (ii) f' (a) and f'…

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Q: Consider the function g : ZxZ → Z × Z defined by g(x, y) = (4xy, 9x + 5y). (a) Is g injective? (b)…

A: (a) we see (5, 0) and (0, 9) are different elements in ZxZ.      But, g(5, 0) = (4.5.0, 9.5+5.0) =…

Q: Determine if the function f : R → Z, defined by f(x) = L] is a bijection. Justify your answer. Show…

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Q: Let f:R-R defined by f(x) =x² , then f(x) is bijective صواب ihi

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Q: Prove that f : R → R by f(x) = 2x+4. Prove that f is one-to-one and maps surjectively onto R.

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Q: Prove that the function f: R W {2} R {5} defined by - 5x + 1 f(x) = #-2 is bijective.

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Q: Prove or disprove whether the function f (r) = a +2 is a bijection from R R %3D

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Q: Let f : A → B be a function. Prove that f is surjective if and only if, for every pair of functions…

A: Please comment if you need any clarification. IF you find my answer useful put thumbs up. Thank you.

Q: Let f : A → B be a function. Prove that if C C A and D C A and f is injective, then f(C) = 7 (D) =→…

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Q: Prove that the function g:R → R* • defined by g(x) = 2* is an injective homomorphism that is not…

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Q: Let f:A at f is surjective. B and g: B -C be two functions. If gof is surjective and g is total…

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Q: f is a bijective function

A: False: Every function has an inverse We have to find out which statement is false.

Q: Let Z4 denote the set of positive integers and consider the function f : Z+ → Z, defined by f(n) =…

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Q: Suppose that f is analytic on a domain D andVz € D,f'(z) = af(z) for some constant a. Show that f(2)…

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Q: Let f : A B be a function. Prove that f is surjective if and only if, for every pair of functions g…

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Q: Let f: {4,5,6} → {1,2,3} be a function defined by f(x) = x-3. Prove that f is injective and…

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Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!