Prove the given theorem and provide (1) examples. Theorem 1.4 (Intermediate Value Theorem) Let f(x) be a continuous in [a, b] and let k be any number between f (a) and f(b). Then there exists a number { in (a, b) such that f() = k. y=f(x) y3Dk f(b); f(a) b Figure 1.1 Intermediate value theorem.

Q: State and prove Rolle's theorem and check the applicability/validity of f(x)=x² at [-1,1]

A: Answer

Q: 1. Let f : Z6 → Z6 be such that f(7) = x2 + 3. (a) Is f a well defined function? (b) Is f…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub parts for…

Q: Let f and g in F[x] be non zero elements. (a) deg(fg) = deg(f) + deg(g) (b) f is a unit iff deg(f) =…

A: For the solution of the problem follow the next steps.

Q: Let a function f be analytic everywhere in a domain D. Suppose that f(z) is pure imaginary for all z…

A: Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: 1.3.2. Let f: X → Y be a function, let B be a subset of Y, and let {B;}ie1 be a family of subsets of…

A: Let f:X→Y be a function, let B be a subset of Y and Bii∈I be a family of subsets of Y.  To prove…

Q: Let f: [2, 7] → [0, ∞) be continuous in [2, 7] a (2,7). Then prove that there exists at least one…

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Q: Prove that if f is a continuous function on a closed interval [a, b], then

A: We have to prove the given inequality for a continuous function on a closed interval [a,b].

Q: Sketch the graph of a function f such that f (x) ≥ 0 on [0, 1] and f (x) ≤ 0 on [1, 2], whose…

A: Consider the function Then and The graph of the function is shown below,

Q: Suppose f X→ Y, and A, B C X. Prove that (a) f(AUB) = ƒ(A) U ƒ(B) (b) ƒ(ANB) ≤ ƒ(A) Ñ ƒ(B)

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Q: g(x) = e2" + x² – 2 with R>0 - 2²

A:

Q: Theorem 7.3. Let X c Y be topological spaces. The inclusion map i : X → Y defined by i(x) 3D х is…

A: Suppose U is an open set in X. Its inverse image is then i−1(U) = {x ∈ A : i(x) ∈ U}          = {x…

Q: Let f : {1, 2, 3} → {a, b, c} be one-one and onto function given by f(1) = a,   f(2) = b and f(3) =…

A: Consider

Q: Derive the autocorrelation function of: x₁ = $xt-1 + εt + 0€t-1 assuming that p < 1, and E(?) = 0²,…

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Q: QUESTION SEVEN МА The diagram below shows the curve with the equation y = 3 + e-'and the %3D…

A: We will find out the required values.

Q: A function f is continuous on [-3, 5] and contains the points (-3, -4), (0, ƒ(0)), and (5, –3). If ƒ…

A:

Q: DETAILS Define F: ZxZ→ Zx Z as follows: For every ordered pair (a, b) of integers, F(a, b) = (2a +…

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Q: Let X = {a, b, c, d,e} and let T = {X, Ø, {a}, {a,d}, {a,e}}, then * O (X,T) is not connected but it…

A: No information is given for topology on Y So i have solved only the middle question

Q: (b) ƒ(AnB) ≤ f(A) Ñ ƒ(B)

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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: THEOREM Let f be a function from A into B. If A, and A2 are subsets of A, then (b) f(A, N A2) C f(A)…

A: We solve this by drawing related picture.

Q: continuous map со

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Q: Prove: There exist functions f, g, h in F such that g o f = h o f, but g is not equal to h.

A: The objective is to prove there exist functions f, g, h in F such that g o f = h o f, but g is not…

Q: 6) Show that the function f:Z → N given by 2n +1, n20 f(x) = -2n, n<0. is 1-1 and onto. I

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Q: f: (R, +) → (R*, .) where f(x) = 2*

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Q: Let f: X --> Y and g, h: Y --> Z be functions. Prove or disprove:  (a) If g of f is 1-1, then f is…

A: (a) Suppose g o f is 1-1. To prove f is 1-1.

Q: f(2) = e=².

A:

Q: If f is a function from X to Y and A, B are subsets of X then .A f(A – B) = f(A) – f(B) .B f(AUB) =…

A: Given: f:X→Y and a,b⊆X, Find out: To choose the correct option.

Q: Let ƒ : R → R be a continuous function such that f(0) 0. Prove that f(1)< 1 for all real x.

A: given f:ℝ→ℝ is a continuous function such that f(0)≤1 for all real x f(x)2-3f(x)+2≥0 to prove f(1)≤1…

Q: 4) Consider the function g(x) = [2r – 1] where g : R → Z. a) Is g a one-to-one function? Explain. b)…

A: