Suppose f: R→→ R is convex. Show that for all x € [a, b]. x-a :) ≤ b = = f(a) + ² = f(b) b a f(x) ≤

Q: (c) For the set A = {[0], [6]} ≤ Z12 determine the set U = h(A). Then for the set UC Z18 determine…

A: In group theory, sets like Z12​ and Z18​ are examples of cyclic groups under addition modulo 12 and…

Q: (e) Use a nullity argument to determine whether f injective. (f) Use a rank argument to determine…

A:

Q: Let f: R³ R be the function defined as f(x, y, z)= √√₂² + y² + 2², 0. if (x, y, z) = Q³, otherwise.…

A:

Q: Suppose that g: [a, b] R is a continuous function. Show that there exists a point c € (a, b) so that…

A:

Q: For S in the domain of f, let f ( 5 ) = { f(x): * € S}. Let C and D f de subsets of the domain of f.…

A: S is in the domain of f fS=fx: x∈S Let C and D be subsets of the domain of f Then we have to show…

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.

A:

Q: Given that there exists a surjective function f : S → T, which of the following substitutions for…

A:

Q: Let f: S¹→ S¹ be a continuous function. Proof or disprove the following statements : a) f has a…

A: Given that the  function f : S1 → S1 be a continuous function  where S1 = x,y∈ℝ2 : x2 + y2  =1 We…

Q: Let f : D → R be defined by f(x) = (x+1)2/x for x ∈ D (D ⊆ R is the largest possible domain of the…

A: The domain of the function  f:   Domain: The domain of a function is the set of input or argument…

Q: 2. Let f: A → B and g: C → D be functions. The product of f and g is the function defined as…

A:

Q: 1. Define f : (- o, 2) → (- 0, 1) by f(x) = . Show that f is bijective. x+1 *-2

A:

Q: f: A → B is a function, Let B1 ⊆ B ve B2 ⊆ B. f−1(B1 ∪ B2) = f−1(B1) ∪ f-1(B2) Show that it is…

A:

Q: Let ƒ : R\{1} →→ R\{1} be defined by f(x) = x/(x - 1). Show that ƒ is bijective and find f-¹. Also,…

A:

Q: 4. Let X an Y be sets with A, B C X and let f: X→Y be a function. Recall that {f(x) EYE A}, and…

A:

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: 18. Let A = {1,2, 3}, B = f: A → B, and g : B → C, such that {W, Y, Z}, C = {7, V2, V5}. Find…

A:

Q: Let f: A → B be a function and let a ∈ A and b ∈ B. Does this mean it is true that f(a) = b? If not,…

A: We use definition of function .

Q: 7. ) Let f: N→ Z with f(n)=n+4. Let A = {1,2,3} and B = {1 € Z \ I <10). Use these to find the…

A:

Q: B. Let f: R²R be a continuous function, and let S be the subset of the R2 given by all pairs (x, y)…

A: Given: Let fx,y=x2+xy+3y2-3. To Show: fx,y contain minimum and maximum value.

Q: a)Suppose f: R → Z, where f(x) = [x + ¹] i. Graph the function f(x). ii. If A = {x: 1 ≤ x ≤ 3}, find…

A:

Q: Given a ring S = S,+ with unity e, a ring Question 3. T =, and an isomorphism : S→ T. (a) Show that…

A:

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: or the given function f(x) = [3x + 5], compute f ({2}).

A: