Let f be a function from X to Y . For A ⊆ X, let also f(A) denote the set f(A) = {f(a) | a ∈ A}. Prove that f is one-to-one if and only if f(A ∩ B) = f(A) ∩ f(B) for any subsets A, B ⊆ X.

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A: fx=lnx,  gx=x3,  1, 4 Both the functions are continuous on 1, 4. Both the functions a re…

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Q: a. If f(x,y) is convex with respect to (x,y) and C is a convex set, show that g(x) = infyec f(x,y)…

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Q: Let a, b ∈ R with a < b and let f : [a, b] → [a, b] be continuous. Show that there exists x0 ∈ [a,…

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Q: Assume that (a, b) is a bounded open interval in R. Using Mean Value Theorem, prove that if f : (a,…

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Q: Verify that the Extended Mean Value Theorem can be applied to the functions f and g on the closed…

A: Consider the provided function, fx=x3  and   gx=x2+1    0,1 Use the mean value theorem and find the…

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Q: If f is a function having as its domain and range the set N of natural numbers, then f is bijective.

A: For the solution follow the next step.

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Q: Give an example of(a) a function f : Z → N that is both one-to-one and onto N; and(b) a function f :…

A: Here, we need to give an example of(a) a function f: Z → N that is both one-to-one and onto N.

Q: Verify that the Extended Mean Value Theorem can be applied to the functions f and g on the closed…

A: Given functions :                          f(x) = 1xg(x) = x2-4         Differentiating f(x) with…

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Q: a. Prove that for any sets A,B⊆X and for any function f:X⟶Y, f(A∩B)⊆f(A)∩f(B).       b. Prove that…

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Q: Use Rolle's Theorem to prove that if f'(x) = 0 for all x in an interval ( a , b ), then f is…

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Q: Let g(x) be a function defined on R such that g''(x)= 1/(1+x^2), g (0) = 0 and g'(0) = 0. Use the…

A: We are given that g''(x)= 1/(1+x2), g (0) = 0 and g'(0) = 0. So, g'(x) = tan-1(x) + c.  But g'(0) =…

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Q: Let A be a nonempty set and let f : A → B be a function. Prove that f is one-to-one if and only if…

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Q: Let F be a "function" defined on the class

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A: Here we use IVP theorem to prove this

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Q: Let f be a function on an interval I and let ₁ and 2 be any two points in I. Then if f (x₁) > f (x₂)…

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Q: Let f : A → B be a function between sets A and B. Let X1, X2 ⊆ A and Y1, Y2 ⊆ B. a) Prove that if…

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Q: Assume f : [a, b] → R is a continuous function on the closed interval [a, b] and differentiable on…

A: Assume f : [a, b] → R is a continuous function on the closed interval [a, b] and differentiable on…