Let f : S → T be a function, and let A and B be subsets of T. Prove or give a counterexample to the statement f-"(An B) = f-'(A) n f-"(B).

Q: is uniformly continuous by definition on the interval (-1,1) . 4 - * Show that f(x) =

A: A function fx defined on an open intervala,b is uniformly continuous on a,b iff  i fx is continuous…

Q: Prove that there does not exist any function f(x, y) such that

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

Q: 34) If f is an even function that is continuous on [-1,3] and if f f(t)dt f₁f(t)dt = 8, then f…

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Q: Verify that the three conditions of the hypothesis of Rolle's Theorem hold for the function f (x) =…

A: 1. f(x) is continuous on the closed interval [a,b] 2. f(x) is differentiable on the open interval…

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: Assume that a continuous function f:R → R is X- and µ-periodic, where A and u are two nonzero…

A: We know that between two rational number lies infinite many irrational number. But function is…

Q: Suppose that f [a, b] → R is strictly increasing, i.e., that if x₁ < x₂ then f(x₁) < f(x₂). Explain…

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Q: Determine if f(x)=, 1+x2 is uniformly continous on the interval I = (-∞, +∞)

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Q: 3. Compute the value of F.dr if F = -yi + xj and C is a semicircle from (-1,1) to (1,-1).

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Q: Show that if f: A-> B is continuous and A is compavt then f(A) is also compact

A: Let , f : A →B be continuous function. Here , A is compact set. We need to prove that , f(A) is also…

Q: Show that f(r) = O(|x – xo|*) and g(x) = o(]x – roP) imply f. g(x) = o(|x – xo]k+i). - -

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Q: Provide a function f, continuous on (0, 1), such that for all r where f'(x) exists, f'(x) > 0 but f…

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Q: Prove that a decreasing function f on [a,b]is integrable.

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Q: Use Cauchy's residue theorem to evaluate the function f(z)

A: Let C is the circle z=2 oriented counter clockwise (in the Positive sense). Then the function…

Q: Define the function f(x, v) = 4x³-x³y-4y5 3x5-3y5 at the origin so that it becomes continuous there.

A: A function is said to be continuous at (a,b) if the the value of function at (a,b) is equal to…

Q: Let (fn) be a sequence of real valued functions defined by cos(x) fn(2) = n By definition, show that…

A: Given  fn(x)=cos (x)n

Q: Let f be differentiable on R. Suppose that f'(r) # 0 for every r. Prove that f has at most one real…

A: Given data is: Let f be differentiable on R. Suppose that f'x≠0 for every x. To prove f has at most…

Q: 3.Show, by means of the example f(t) = sin t, that f ∗ f is not necessarily nonnegative.

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Q: Describe the long run behavior of f(z) = z" As z + - 00, f(z) - As z + 0, f(z) → ( -0-

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Q: Let f(x) = x -[r] and f :[1,5) →[0,1). What is the discontinuously type of f(x) at the point x = 3?

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Q: Let f(æ) = In(z') f'(2) =

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Q: Can Rolle's Theorem be applied to f(x) = 1/(1-x) on interval [0,3]? If yes, what is c in (0,3) such…

A: here we can't apply the rolles Theorem

Q: 4. Let f be defined for all real x, and suppose that \fx) – f(v)| < (x – y)? for all x, y e R. Prove…

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Q: Assume that f is a function with domain [0,1] that is differentiable in (0,1). Assuming that f (0) =…

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Q: Let f(x) be twice differentiable and invertible function such that f'(x) >0 and f"(x) > 0, prove х,…

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Q: Suppose f and g are differentiable functions on R such that,

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Q: Let f be differentiable on R. Suppose that f'(x) # 0 for every r. Prove that f has at most one real…

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Q: Let f (x, y) = 4-y² In(y–z²) %3D Draw a picture of the domain of this function.

A: x,y ∈R2

Q: Suppose that u(x) is a differentiable function such that

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Q: Differentiate f  and find the domain of f .   f(x)   = In  In  In x

A: we have to differentiate f and also we have to find the domain of f  f(x)   = In  In  In x

Q: Suppose that fn ⇒ f on R and that each fn is a bounded function. Prove that f is bounded.

A: Solution

Q: Find a function flx) such that f(x) -llex and %3D 3. f(0)-- 주

A:

Q: Prove that f (x) = x2 takes on the value 0.5 in the interval [0, 1].

A: Here given

Q: Let f: X→Y be a function. Please prove that if f(X-A)=Y-f(A) for any A⊆X, then f is onto.

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Q: Let f be a function defined on [a,b] ,if f is increasing ,then f is continous   True or false

A: True

Q: Use the definition to prove that the function in f(x) = 2x - 3 is uniformly continuous.

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Q: Determine whether the given set of functions are linearly independent or linearly dependent on the…

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Q: Differentiate the function.

A: Given function -f(u) = lnu1+ln(2u)Differentiating with respect to u - f'(u) = ddulnu1+ln2uf'(u) =…

Q: If f is increasing on the interval [a, b), then f' does exist. Select one: a. True b. False

A: Let f is increasing on the interval [a, b]. Since, there is not given that the function is…

Q: 5. Show that the functions x,x? -,x³ -x x are mutually orthogonal in the interval [-1,1].

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Q: Let f = f(x, y), x = s +t and y = s - t. Show that (9- ) - se se as at ax ду

A: It is given that, f=f(x,y), x=s+t and y=s-t. Show that, ∂f∂x2−∂f∂y2=∂f∂s·∂f∂t.…

Q: Suppose f is a bounded real function on [a, b), and f' eR on [a, b). Does it follow that fe R? Does…

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Q: Assume that f'' exists and f'' (x) = 0 for all x. Prove that f (x) = mx + b, where m = f'(0) and b =…

A: Given that  f'' exists and f'' (x) = 0 for all x. Integrating both sides ,we get By substituting…

Q: If f : R + Ris continuous and periodic, prove that f is uniformly continuous.

A: Given f:R→R is a periodic function with period k that is f(x+k)=f(x) To prove: if f:R→R is…

Q: Let S(x²+2x)², x> 1, f(x) = Tax+ bx², ах x< 1. Compute for all a and b such that the function f(x)…

A: Proceed as shown

Q: Suppose that a function f(x, y) satisfies fxx fyy – (fxy)² > 0 at a point (a, b). How is the sign of…

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Q: Prove that the function f : R → R given by f(x) = 3x° – 2x is continuous at 1 using the a.…

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