Let f : A → B be a function between sets A and B. Let X1, X2 ⊆ A and Y1, Y2 ⊆ B. a) Prove that if X1 ⊆ X2, then f(X1) ⊆ f(X2). b) Prove that if Y1 ⊆ Y2, then f -1 (Y1) ⊆ f -1(Y2)

Q: f,(x)= e* f2(x)=e¯* (x) = e'

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A: To show that there does not exist any function that satisfies the given conditions

Q: 7. Prove that the function f:Z → Z defined by f (x) = x + 8 is one-to-one and onto.

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

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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.

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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…

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Q: Consider the functionf defined by f(x,y) = k. Use the definition to show that for fixed real numbers…

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Q: If  and f is continuous, then  f must have at least one zero between  a and b. (Assume a ≠ b.)

A: Case1: Whenfx=0 , then ∫abfxdx=0 is obvious.

Q: f(ix, λγ) - λκ. f(x, y)

A: Given: fλx,λy=λkfx,y

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: Suppose that f'(x) > M for all x E [a, b]. Prove that f(b) > f(a) + M(b

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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.…

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Q: Let f be differentiable on the closed interval [a, b] such that f′(a) = y1 and f′(b) = y2. If d lies…

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Q: Suppose f and g are continuous on a closed interval [a, b] such that f(a) > g(a) and f(b) < g(b).…

A: I have used intermidiate value theoram of continuous function.

Q: Let f be a bounded measurable function on a set E of finite measure. For a measurable subset A of E,…

A: Below given clear step by step explanation

Q: Prove that f (x) = x-12 is : (i) not sectionally continuous in every finite interval in the

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Q: Prove that f (z) =zz is nowhere analytic

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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: 3. If f:(X,r)(Y.A) and g:(Y,A)→(Z,r) are open functions, prove that gof :(x.r)(Z.r) is open…

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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: Let I be an interval and f : I → R be a function with Darboux property such that f(I) is finite.…

A: Let I be an interval and f : I → R be a function with Darboux propertysuch that f(I) is finite.…

Q: f(c) = g(c).

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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. Show that the function f : R → R* defined by f(x) is not onto. x2 +1

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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: et f: (a, b) → R. If f'(x) exists for each x, a < x <b, prove that f is continuous there.

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Q: The function on ℤ defined by f(x)=-x^2-5 is onto. True or False ?

A: The given function is                  f(x) =x2-5   Graph:

Q: Let f : [0, 1] → R be a function with Darboux’s property such that for any y ∈ R, the set  f…

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Q: Suppose that f1 : [0, 1] → R and f2 : [0, 1] → R are continuous everywhere and that f1(0) f2(1).…

A: Intermediate value theorem : Suppose be a continuous function then it takes all values between .

Q: If f is a function satisfying |f (x) – f(y)| < (x – y)? - for all x and y, prove that f is constant.

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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: Suppose f : [a,b] → R is a continuous function. Prove that the direct image f([a,b]) is a closed and…

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Q: Let fi and f2 be concave functions from R" to R. Let f : R" +R be defined as the pointwise minimum…

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Q: Prove that if f is a continuous function with domain 0, 1] and X is the set of all x in 0, 1] such…

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Q: If f is a function of only r, g is a function of only y, and f and g satisfy In(f(x)) + In(g(y)) =…

A: Given: f is a function of the only x and g is a function of the only y.…

Q: Show that the functions(a) f(x)=1/(1+x2),x∈R, (b) g(x)=x/(1+x2),x∈R, are uniformly continuous, while…

A: Answer