Consider a function f : D → R, where D C R. Suppose the function satisfies the property f(xy) = f(x) + f(y) and that f(x) is not zero for all x € D. (a) By choosing suitable values of x and y, and making no other assumptions, show that domain D cannot contain 0 and that f(1) = 0. (b) Assuming that f is differentiable, find f'(x) in terms of f'(1). Hence show that dt [² t/t . t 1 f(x) = f'(1)

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A: Use Hessian matrix

Q: Let f : R R:f(x) = 10x + 7. Find the function g : R→R such that gof = fog = IR. %3D %3D

A: Given f: R -> R  such that f(x) = 10x + 7 and  g : R -> R such that g o f = f o g = IR

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A:

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A:

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A:

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A:

Q: Let f: X→Y be a function, and let A,BCY. Prove or disprove: f-¹(A)uf-¹(B).

A: The given statement : f-1(A ∪ B) = f-1(A) ∪ f-1(B) is true and we shall prove this statement.

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A: Properties of Laplace transformation

Q: 9: Check Euler's Theorem for the Function

A: We checked Euler's theorem for function.

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A: we have to find the domain for f(x)

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Q: Show that the following relations are not functions on R. a) ƒ = {(x, y) = RxR | y=x/(x² −7x+12)}

A: Given function. y=xx2-7x+12 is not define for x2-7x+12=0 =&gt; (x-3)(x-4)=0

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 -…

A:

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A:

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Q: n For each nEN, consider x = with x ER+. Define the sequence {gn} by x (a) (b) A = [1,2) CR. (1)…

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Q: Let f(x) = 5- x and g(x) 2, find: a. (fog)(0) b. (go f)(0) = c. (fo g)(x) Preview d. (go f)(x) =…

A: Value of the above ques is

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A:

Q: B. Let f: R²R be a continuous function, and let S be the subset of the R2 given by all pairs (x, y)…

A: Given: Let fx,y=x2+xy+3y2-3. To Show: fx,y contain minimum and maximum value.

Q: 3) Show that f = f, ху for the following function: ху (x° у) f(x. y) + In(x° + y30 ) 20

A: We have to prove that fxy = fyx

Q: Consider the function ƒ : R → R defined by 2x + 5, ax² + x, Show that there is a value of a for…

A: