Let ƒ : R → R, x ↔ x² − 4x + 7. (a) Is f injective? If not, find the largest possible set DCR such that ƒ : D → R is injective. (b) Is f surjective? If not, find the largest possible set CCR such that ƒ : R → C is surjective. (c) Is f bijective? If not, find D, C C R such that ƒ : D → C is bijective. Find the inverse function f-¹: C → D. (d) Draw the graph of both functions, f: D→ C and f-¹: C → D.

Q: Discrete math.  Let S = R. On S we define the relation R as follows (∀x)(∀y)(xRy ⇔ y ≥ x + 3). Is R…

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Q: Let F= R and V = R*. Let W, = {(x1,x2,X3,X4) € R* : x1 – x2 + X3 – x4 = 0 and x1 +x2 = 0}, W2 =…

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Q: f: X→ Y and g: Y→V be bijective functions. Show the following: ) gof: X→V is injective. ) gof: X→ V…

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Q: b. For each of the following, determine whether f is a bijection or not. If so, prove it. If not,…

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Q: (-х+а, х < —1 If f(x) = {x² + 5, –1 < x < 2 is continuous then х — b, х 2 2 a + b =?

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Q: Minimize the function ƒ : R² → R given by f(x1,x2) = x² + 2×₁×2 + x² on the set X = {(x1, x2) Є R² :…

A: The objective of this question is to find the minimum value of the function f(x1,x2) = x1^2 + 2x1x2…

Q: f: X→Y and g: Y→V be bijective functions. Show the following: ) gof: X→ V is injective. ) gof: X → V…

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Q: S and T are relations on the real numbers and are defined as follows: S = {(x, y)| x y} %3D What is…

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Q: Minimize the function ƒ : R² → R given by f(x1,x2) = x² + 2×₁×2 + x² on the set X = {(x1, x2) Є R² :…

A: The function is minimum at critical point (x1​,x2​)=(21​,21​)and the minimum value is…

Q: Q: Show that the functi on f:R →R given by f (x) = ax +b, where ab e R, a #0 is a bijection.

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Q: Suppose UCC is a domain, f, g = O(U). If |ƒ(z)|2 + |g(z)|² = 1 for all ZEU, show that ƒ and g are…

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A: Given relation is ρ =  a, b ∈ ℝ × ℝ : 1 + ab &gt; 0

Q: How many functions f: {1,2,3,4} {1,2,3,4} are there satisfying: a. f(1) = 1 or f(2)= 2 (or both)? b.…

A: Solution :-

Q: 1]If 9:T>V and f:ST and gof IS injective, prove that both f añd 9 are injective.

A: Injective function:- for every x1=x2 implies that f(x1)= f(x2) .or vice versa.

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Q: If f A —> B and g : B —> C are injective function, then gof: A —> C is an injective function. Prove…

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Q: Let X and Y be sets with A,B ⊆ X. If f : X →Y is a function, prove that f(A∪B) =f(A)∪f(B)

A: I am going to solve the problem by using some simple algebra to get the required result of the given…

Q: Prove the statement. Let f : A → B . If X ⊆ A, Y ⊆ B , and f-1[Y] = {x ∈ A|f(x) ∈ Y}, then f[X] ∩ Y…

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Q: Exercise.) Prove that if f is a real function on a measurable space X such that { | f(x) 2 r} is…

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Q: If f(x, y) = 2√/16 - x² - y², then the domain of f is the set D = {(x,y)|x² + y² ≤ 16}. Select one:…

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Q: c) Let the set A = {-1,0,1} and consider the following function: f3 A × A → A × A given by f(x, y) =…

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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: 1. Consider the functions f : A → B and g : B → C. a. If f and g are both injective, show that so is…

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

A: Using derivative

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A: Unity: Let R, +, ∘ be a ring. If the semi-group R, ∘ has identity element then it is called as unity…

Q: Prove: If f₁, f₂ € R(a) on [a, b] with f₁ ≤ f₂, then so f₁ da ≤ f f₂ da.

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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: The restrictions on the domain for f (z) = 2x + are +2 O True O False

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Q: Let f = u + iv be analytic in the connected open set and assume u and v satisfy u? + 3v2 = 4. Show f…

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