1. Exactly one of the following requests is impossible. Decide which it is, and provide examples for the other three. In each case, assume f,g: R→ R are functions. (a) Functions f and g which are not differentiable at zero but where fg is differentiable at zero. (b) A function f not differentiable at zero and a function g differentiable at zero where fg is differentiable at zero. (c) A function f not differentiable at zero and a function g differentiable at zero where f + g is differentiable at zero. (d) A function f differentiable at 0 but not differentiable at any other point.

Q: In(1 – x²) Vy – Vy? – 1 V1 - x2 (a) f(x, y) = (b) ƒ (x, y) = %3D

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Q: В. B. Show that the function f (z) = -i z2 is analytic.

A: The solution is given as

Q: Show that f[x0, x1, x2] = f[x2, x1, xo].

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Q: 3. Determine the domain of the following function, then sketch it: 1 f(x, y) = V² + y? – 9

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Q: The function f(z) is analytic if

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Q: 5. Prove that the function f: Z → Z defined by f(x) = 3x + 1 is one-to-one.

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Q: Determine whether the statement " If f is a function, then f(ax, ay) = a2f(x, y)" ,is true or false.…

A: If f is a function, then f(ax, ay) = a2f(x, y).

Q: If the function f: R\{2} → R\{2} defined by f (x) = 2х+3 then the composite function х-2 fof (x) =.…

A: Use the definition of composite function f∘gx=fgx, to obtain f∘fx. fx=2x+3x-2f∘fx=ffx=f2x+3x-2

Q: Let function f(x) be defined ás f(x) = x² + bx + c, where b, c are real numbers and f(1) - 2f(5) +…

A: Clues : Let the function f(x) be defined as f (x)  = x²+bx+c where b, c are real numbers and f(₁)…

Q: Show that R(g ◦ f) ⊆ R(g). (b)  Give an example of two functions where R(g ◦ f) ⊂ R(g). Briefly…

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Q: [9] Let A = {a, b, c, d} and B = {1, 2, 3}. Define a function F: A → B as follows: F3(а, 3), (b, 3),…

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Q: Given f: R → R where f(x) = x³ Is it a function or not? If function, identify whether it is…

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Q: Let f(z) = , then: a. f is analytic function everywhere except z = 1. b. at z = 1,f is analytic…

A: Given Information : fz = 11-z

Q: Examine whether the following function is Analytic or not f(z) = [z]²

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Q: 1.) fex) =x3 -x²+4x +2 2.) FCxS= -x3+ Sx-2 %3D

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Q: g: RR, g(x) = x² − x − 6. - - Express you answer as an interval or a union of intervals.

A: Given: g(x)=x2-x-6 We know that for a parabola given by the equation f(x)=ax2+bx+c , the coordinates…

Q: C. State whether or not the following relations are functions. Also, give their domain and range. 1)…

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Q: Consider the function f : A → B.

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Q: True or False: The antiderivative of a function is uniquely defined, that means there is exactly one…

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A: f(t) = t/(1-t)2 domain means input value which we can input in function and function is valid f'(t)…

Q: Let f : A → B. Then 1âf =ƒ and ƒ1a= f. Hint. You may use the fact that fg, for functions f, g, means…

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