Show that the operator K: L²[0, 1] → L²[0, 1] given by K f(x) = f f(y) √x+y x = [0, 1] is compact.

Q: Let f(x, y) = 125-3x²-2y2² Evaluate f(0, 3)

A: f(x,y)=sqrt(125-3x2-2y2)Find f(0,3)

Q: 3. Find the fixed point(s) of f : [1, 0) → [2, 0) defined by f(x) = vx² + x.

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Q: Show that the function f(x) = 2x³ – 1, from f: R → R is bijective and find its inverse.

A: The function f(x)=2x3-1, from f:ℝ→ℝ

Q: If f(x) = x + /2 – x and g(u) = u + v2 – u , is it true that f = g?

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Q: Determine the largest subset S of C where the function f(z) = 3x2 + 2x - 3y2 - 1 + i(6xy + 2y) is…

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Q: 3. (b) Consider the function f(x, y) = a(x² + y²) + xy + 1 for some real number a. Find Vf. Find fxx…

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Q: Let f : X → Y and g : X → Z be two bijective functions. Is the function h: X → Y × Z defined by h(x)…

A: Given: f:X→Y and  g:Y→Z be two bijective functions. To Check: Whether the function h:X→Y×Z defined…

Q: Find the adjoint of F:C3 tends to C3 defined by F(z1, z2, z3)=(2z1+(1-i)z2 , (3+2i)z1 -4iz3 ,…

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Q: Find and sketch the domain of z = f(x, y) = In(9 - x - y²).

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Q: . Show that the function f: R→ R defined as f(x) = 13x − 15 is one-to-one.

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Q: anction of z = x + iy

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Q: = Evaluate S₁2₁,22,23] f (2) dz where 21 = −i, z2 = 2 + 5i, zg 5i and f(x + y) = x² + iy

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Q: 2. Sketch the domain of the following functions: a) f(x, y) = b) f(x, y, z)= 2-x-y - eª %2 In(4-x² -…

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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: Draw a graph and use geometric arguments to determine the value of lim RN. N→∞o

A: Let's consider the Riemann sum for the function f(x)=3x+1 on the interval [ 1,3]. If we divide the…

Q: Show that the function f(x)=√x+1 is differentiable in 2 = 2 but not differentiable in x = −1.

A: Given data:- The given function is, fx=x+1

Q: The domain of the function f (x, y) = In(2 – /x² + y² ) is:

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Q: Find f, and f, for f(x,y)=y In (3x +4y). fx fy

A: When the partial differentiation is done with respect to a specific variable, then all other…

Q: 2. Let f= L(X,Y). Show that ƒ is one to one iff (f(x).(x)....(x)) is linear independent when (x,x)…

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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: Select the following functions that are bijective. ƒ₂: R → R, ƒ₂(x) = 14 f₂: R → R, f₂(x) = x f₁: R…

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Q: 3x +1 III.(10) Prove that the function f: R – {5} –> R – {3} defined by f(x) = is a bijection. x- 5

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Q: Let f(x) = x² and let P = { -3, -5/2, -2, -3/2, -1, -1/2, 0 }. Compute Lf (P) and Uf (P). Lf (P) =…

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Q: Let F = (2xz+y², 2xy, x²+1), if Vf= F, find f(1, 1, 1) – f(0, 0, 0)

A: Given vector funtion and

Q: Find the values of x for which the gradient of the tangent to the graph of y=f(x) is equal to the…

A: We have to find the values of x for which the gradient of the tangent to the graph of y=f(x) is…

Q: Consider the function f: N× N → Q given by f(x, y) = your claim. X y + 1 Find the image of f and…

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