Let f: X → Y and g: Y → V be functions. Show the following: (a) f is injective 3h : Y→ X such that ho f = id, (b) f is surjective 3h: Y→X such that fo h = idy Use: v injective if x1 xn → v(x1) ‡ v(xn) # How to prove these Ihs) Can you explain id x How to proof with the arrows (do you prove both rhs and

Q: 2x - 3x + 13x+a if x#2 f(x) = . X-2 b if x = 2

A: Solution:- It is give that f(x) is continuous at x=2  and the given function is  f(x)=…

Q: Suppose f:R→IR is a diffexetiuble function which satısfies flo) = 0 . We wort to shouw f': IR → IR…

A: We have to do implicit differentiation.

Q: Suppose 2 < f'(x) < 6, for –1 < x< 1. Then f(1) – f(-1) is between A) 2 and 10. B) 4 and 12. C) 0…

A: GIVEN : 2 ≤f'(x) ≤6,  for -1 ≤x≤1 TO FIND : f(1) - f(-1) HINT : Mean value theorem

Q: 5) Let f be differentiable at a. Show that f(z) is differentiable at a if and only if f' (a) = 0.

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A: Since x and y are arbitrary points  and f'exists. hence F is continuous and also differentiable. let…

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Q: If f is a function from X to Y and C, D are subsets of Y then A O f(C-D) =f(C)-f (D) B f'(Cn…

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Q: Are these functions injective or surjective?  f : R → R; f(x) = x^2,  f : R → Z; f(x) = bxc,  f :…

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Q: Let z be a differentiable function of x and y related by the equation: x z ++* =(√√y+cosh(z))". X 2…

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Q: 1 is IR? \ {(0,0)}. 22 + y2 The domain of the function f(*, y) True O False

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Q: True f must be continuous in its domain? or false: If f(x) is differentiable in its domain then

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Q: a) f(x,y) = x³ + y³ + y(3x²y+ 3ry²) b) f(x,y) = x² + iy³

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Q: Let /: X→Y be a function, A, B C X and C, DCY. Show that: 3. If CSD then SF (D) 4. f¹(COD)=f-¹(C)…

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Q: Check that the equation x³ + y³ + z³ = x + y + z implicitly defines at least one differentiable…

A: Solution

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Q: show that the following function analytic f(z) = 4e²+ Sin Z arp

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A: Given function is y=cosh(ln6x3)

Q: 3. Let f be a bounded continuous function on C and a E C. Show that f(z) dz lim 0. iz]=r (z – a)²…

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Q: Show that w=f(z) =3ix+y dose not satisfy C.R.E. thus it is not analytic function

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