Q3: a) Let f : Z → Z be such that f(x) = Vx² – x + 1. If invertible, and if it is, what is its - inverse?

Q: 0 0 0 1 0 0 0 1 0 ) Let A =

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Q: (a) Let f(x) = tanx- 2, g(x) = x5 + x³ and h(x) = Vcsc5x. Then find %3D the f(g(h(x))), using chain…

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Q: af Q1: a) Use the chain rule to calculate for the function f(x, y, z) = 3x² - 3y + 4xz², x =…

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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: Define fi z-> 2 xZ by Ala)l (2,72t1) if xis even 18,0) if t is odd a) Is f injective 2? b) Is f…

A: We have f:Z→z×z Such that fx=x,x+1     if x is evenx,0          if x is odd

Q: Find the linearization L(x) at x= a. f(x) = x+ a = - 1

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Q: 3. (a) Show all functions satisfying u"(x) = 0 are linear by integrating the expression of u"(x) %3D…

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Q: Two functions u and v of x and y are said to satisfy the Cauchy- Riemann eguations if u, = v, and u,…

A: Thje given functions are,u(x,y)=y(x2+y2) , v(x,y)=x(x2+y2)

Q: Y Show that f (2)= 2+ Con formal maps %3D on %3D

A: Given that,f(z)=z+1zz=1

Q: Let f be the function from {a, b, c} to {1, 2, 3} such that f (a) = 2, f (b) = 3, and f (c) = 1. Is…

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Q: please solve this linear algebra problem.

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Q: Let D, be the linear transformation from C'[a, b] into C[a, b]. Find the preimage of the function.…

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Q: 3. Let f: A-→ B and g : B A be functions. Suppose that y = f (x) if and only if r =g (y). Prove that…

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Q: 2) Use the Wronskian to show the three functions are linearly independent. f(x) = x, g(x) = cosx,…

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Q: a) Let f: Z Z be such that f(x) 2*x. Is f invertible, and if so, what is its inverse?

A: # we are entitled to solve one question at a time, please resubmit the other question if you wish to…

Q: 1 14. If f is integrable on [a, b), then f is also integrable en [a, b].

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Q: 2. (a) Let f be defined for all x, y by f(x, y) x-y - x. Show that f is concave (i) by using Theorem…

A: Given: a) f(x, y) =x - y - x2 Theorem 2.3.1: This theorem says that if f is twice continuously…

Q: Find the linearization L(x) of f(x) at x=a f(x)= x+1/x a=3 show all work

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Q: fx) =x+x, aニー Lcx) = ? (メフフ %3D

A: Given that fx=x+1x,      a=−1

Q: Exercise 3. Show that f : Z → N so that f (z) = z² + 1 for all z E Z is neither one-to-one %3D nor…

A: One-One Function: A function is said to be One-One if for every element in the domain there exists…

Q: 3. a) Prove that the function sin is not continnous on b) Show that the functwoin g(x) x e RiQ is…

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Q: Suppose that f: R-Cis a measurable function with 1+ |xf° for almost all x E R. Show that its Fourier…

A: Let, f⏜ be Fourier transform of fx. Then nth derivative of f⏜ is Fourier transform of…

Q: Let f(2) := 25. Show that f is invertible in a neighbourhood of i and calculate (f-1Y(/(i)).

A: Let f(z) = z5, show that f is invertible in a neighbourhood of i and calculate(f-1)' f(i)

Q: Find f. f "(x) = x-2, x > 0, f(1) = 0, f(6) = 0 %3D %3D In 6 (In 6) f(x) = -In(u) + u 5

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Q: Prove that the function : R-{1} →R- {2} defined by 2a f(x)= s bijective. 2 1

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Q: Let A: C [0,1]→C [0,1]. Prove that operator continuous reversible, find A. 2.11. Ax(t)= x(1)- -…

A: The given operator A such that A:C0,1→C0,1. We use the inverse operator method for solving linear…

Q: Q3: a) Let ƒ : Z → Z be such that f(x) = Vx² – x + 1. If invertible, and if it is, what is its

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Q: 2. Use the Squeeze Theorem to show that f(r) = 2 is in R(0, 3).

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Q: Let D, be the linear transformation from C'(a, b] into C[a, b]. Find the preimage of the function.…

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Q: Let A: C [0,1]→C [0,1]. Prove that operator A is continuous reversible, find A!, 2.11. Ax(1) = x(1)…

A: We have given operator A such that A: C[0,1]-C[0,1].Also we must know that the inverse operator…

Q: 11. The function f(x) is defined on [1, o0) and satisfies VTO dt. 10 Find f(x).

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Q: ve that the Bessel functionm generating funchion ie too

A: The aim is to show that∑n=-∞∞Jn(x)tn=ex2t-1t The approach for this proof is, Take,…

Q: Let f.9: E1-00, +00] be two integrable functions. Using the fact for non-negative functions, prove…

A: Given that f,g:E→−∞,∞ be two integrable  and no-negative function. We have to prove that…

Q: True or false? If det(A)= 0, then A is not invertible.

A: Explanation: Case 1: If det(A) is invertible. In this case, det(A) not equal to 0 because in matrix…

Q: Exercise 4.12 Give an cxample where f(x) - b as x → a and g(y) + c as y – b but g(f(x)) + c as x → a

A: To find an example where fx→b as x→a and gy→c as y→b but gfx→c as x→a. Let b∈ℝ be a fixed real…

Q: Let A: C [0,1]→C [0,1]. Prove that operator A is continuous reversible, find A. Ax() = x() + ja-m}xt…

A: Solution:- We have given operator A such that A: C[0,1]- C[0, 1] Also we must know that the inverse…

Q: 2-Express each function in the form u(x, y) + iv(x, y), where u and v are real, (c) ez2 (d) In(1+…

A: We have to write in u(x,y) and v(x,y) form.

Q: Example H-W:_for f(x,y) = cos(xy) – x³ + y* compute fxyy, fxxyy %3D

A: We have to find the partial derivative

Q: Let C[0, 1] have inner product [f(x), g(x)] = ∫ f(x)g(x) dx. Find α so that (x + αx2 ) ⊥ (x + 1)

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Q: ) Consider the function f(r) = f(1) has a critical number A %3D f"(A) = Thus we conclude that f(x)…

A: Critical point f(x) is the number at which first derivative is zero.…

Q: Let {yk } be the DFT of {x; }, where x = [0, –1, 0, 1] y3 = O i -1 -i 1

A: Here, x0=0, x1=−1, x2=0 and x3=1 We have to find y3 Now, y3 is given by…

Q: (3x² -1, x 1, Determine c and d so that f(x) is continuous everywhere.

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Q: 2-3. If f is integrable on [a, b], then ("f(x) dx = lim E f(x) Ax i=1 • In the theorem above, define…

A: We know that integration is equivalent to summation as mentioned above. Here, △x is the small step…

Q: Exercise 6: Consider the function: f(x) = arctan Show that f'(r) = 1/(1+ x²). Show there does not…

A: We are given the function,

Q: yLet f(2) be analytic in {z: |2| < 1} with f(0) that |f"(0)| < 2. = 0 and |f(2)| < 1 for all z in…

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Q: 2. (a) Determine whether f : Zx Z → Z defined by f(m, n) = = 2m + n is one-one, onto, both or…

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Q: 4. Show that the function f : X → Y defined by f(X) = X U {a} vX €x is a bijection.

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Q: 5) Let z = f(x2 - y²). Show that y = 0 ду %3D ax

A: Here, ∂ represents the partial derivative, which means when differentiating with respect to x,…

Q: 16. Consider the function f:(-2, 0) → [0, 0) defined by f (x) = vx +2. Prove that f is injective.

A: f(x) here is injective if and only if for all a,b∈(-2, ∞) we have a=b⇔f(a)=f(b)