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

Q: Let f : R R with f(x) = -2x +5 Prove that f is onto.

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Q: Determine whether or not the function f : Z × (Z − {0}) → Z is onto, if f((m, n)) = ⌊mn ⌋.

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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: Prove that f: R R where f (x) = |x| is neither injective surjective. nor

A: To prove this theorem, we use the defination of one one and onto.

Q: Let f : R → R be f(r) = + 1+ \|° (a) Is ƒ everywhere defined? If not, give the domaim (b) Is ƒ onto?…

A: Given, f: R→R be                       f(x)=x1+|x|

Q: Show that the function f : R → R,defined as f(x) = x2 , is neither one-one nor onto.

A: We will show this by counter examples.Let x= 2 and -2 Then f(2)= 2^2 = 4 and f(-2)= (-2)^2 = 4 This…

Q: Let ƒ : A → B be a function. Prove that f is onto if and only if there exists a function g : B → A…

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Q: ) Find a function f(x) satisfying these conditi

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Q: Let f : [a,b] → [a,b] be continuous. Prove that there exists x ∈ [a,b] such that f(x) = x.

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Q: Prove that there is an additive function T : R --> R that is not linear.

A: Let T: V→W  be the linear transformation and we it is linear,…

Q: Let f: R (-1, 1) be a function defined by Prove that (a) f is injective; (b) f is surjective. f(x) =…

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Q: Prove the following statement: For any function f : R" - R and any function g : R" R, let h: R" R be…

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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: 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: Let f be the function from {a, b, c, d} to {1, 2, 3} defined by f(a) = 3, f(b) = 2, f(c) = 1, and…

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Q: Let the function f: R R be defined by f(r) = 3r -5. Prove that the function is bijective.

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Q: T is a function?

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Q: Define a bijection function f : N → {n : n > 10, n e N}, and prove it.

A: We have to define a bijection function f:N→n:n≥10,n∈N and prove it. Define a function…

Q: Consider the function f : R³ → R defined as V(x1, x2, x3) E R³ f(x1, x2, X3) = X1X2+ X1X3 + X2X3 a.…

A: We are given the function f: ℝ3→ℝ defined as ∀x1, x2, x3∈ℝ3    fx1, x2, x3=x1x2+x1x3+x2x3. For (a),…

Q: Show that the function f : R→ R : f(x) = x° is one-one and onto.

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Q: Let L be a linear operator on R1 and let a = L (1). Show that L (x) = ax for all x ∈ R1.

A: It's a highly simple question.

Q: The function f : R → R defined by f(x) = cos(x – –) is a. one-to-one and onto. b. onto but not…

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Q: Let f:A→B. Prove that if X⊆A and f is injective, then f(A−X) =f(A)−f(X)

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Q: Show that the function f: Real #s to Real #s defined by f(x)=x2 is not a homomorphism.

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Q: Prove that if f is continuous in [a, b] and c is the infimum of f in [a, b], then there exists an xo…

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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: (Ь) Give explicit, concrete definitions for two functions f1, f2 : Z → z+ such that: i. f2 is onto…

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Q: Define f: R → [2, 0) by f(x) = x² +2,Vx € R.Prove that f is onto.

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Q: Let f be a differentiable real function defined in (a, b). Prove that f is convex if and only if f'…

A: A function f is convex in a domain a,b if for all 0≤λ≤1, fλx1+(1-λ)x2≤λfx1+1-λfx2  ∀x1,x2∈a,b. Also…

Q: Prove or disprove the relation C on Z defined by C = {(x, y): x=y (mod 5)} satisfies the two…

A: I just used the definition of congruence and divisibility

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: Let f: X→Y be a function. Please prove that if f(X-A)=Y-f(A) for any A⊆X, then f is onto.

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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 be a function from A to B . Define a relation R by (x,y) ∈ R If and only if  f(x) =f(y) .Prove…

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Q: The function f : R → R defined by f(x) = cos(x – 5) is а. one-to-one but not onto. b. onto but not…

A: Let's find.

Q: Define the function f:R → R by f(x) = x³ + 1, Vx € R. a) Prove that f is onto? b) Prove that f is…

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Q: 3. Consider the function f: IR→ (0,1] defined as f(x) = x² +1 (a) Prove that f is not one-to-one.…

A: Given function is f: ℝ → (0, 1] such that f(x) = 1x2+1

Q: 2. Let f: R → R_be f(x) = 1+ |r|" (a) Is f everywhere defined? If not, give the domain. (b) Is f…

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Q: -B-Let f:( M, (Z).+-)-(M, (Z). +..) be a function define by such that g[]=[ EM₂(2). Is f is…

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Q: Consider a function space of the set of functions f(x) of a real variable x ∈ (−∞, +∞), such that…

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Q: Exercise 11: Suppose that f and g are continuous on a, b, and f(a) g(b). Prove that there exists c…

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Q: Which of the following function f: Z XZ → Z is not onto? f(a, b) = a O f(a, b) = |b| f(a, b) = a - b…

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Q: Let f : R → R be the function defined by 1 f(x) 2* a) Is ƒ is one-to-one? If yes, then explain why.…

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Q: Let A be a nonempty set and let f : A→ A be a function. Prove that if f o f = i4, then f is…

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Q: Define the function f: R→ R by f(x) = x5 – 1, Vx € R. a) Prove that f is onto? b) Prove that f is…

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Q: Prove that f is an onto function. Is f an invertible function? If so, find the inverse function of…

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Q: Determine whether or not the function f : Z × (Z − {0}) → Z is onto, if f((m, n)) = ⌊m/n ⌋

A: Given that: f(m,n)=mn,  f:Zx(Z-{0})→Z

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: 3 10. Prove the function f : R- {1}- R-{1} defined by f(x) = ()° is bijective. %3D |

A: Given that the function f:ℝ-1→ℝ-1 is defined by fx=x+1x-13

Q: Consider the function f: Z x Z → Z defined by f (m, n) = 2m – 2n for all (m, n) E Z × Z. (a) Is f…

A: A function f: A→B is said to be one-one if        f(x1)=f(x2)   ∀x1, x2∈A         ⇒x1=x2 In words,…