1. Define f : (- o, 2) → (- 0, 1) by f(x) = . Show that f is bijective. x+1 *-2

Q: Examine the continuity of f(z) = 2-√z+4 Z at z = 0

A: Given: fz=2-z+4z We have to find the continuity of f(z) at z=0.

Q: 5. Determine whether the function f : R+ → R defined by the rule f (x) = positive real numbers x in…

A: A function is said to be bijective if the function is both one one and onto. The given function is…

Q: f,(x)= e* f2(x)=e¯* (x) = e'

A:

Q: 2. Prove that the function f : R? → IR defined by f(x, y) = 1- a2 – 4y? is neither injective nor…

A: Given the function f:ℝ2→ℝ defined by fx,y=1-x2-4y2

Q: Prove that there does not exist any function f(x, y) such that

A: To show that there does not exist any function that satisfies the given conditions

Q: 7- The function f(z) = z is: a) Analytic on C b) Analytic only at zo = 0 c) Not analytic

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Q: 7. Prove that the function f:Z → Z defined by f (x) = x + 8 is one-to-one and onto.

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Q: Let f : R → R be the function defined as if x 1. Use Theorem 32.5 to prove that f is Darboux…

A: A function is Darboux integrable if for all ε>0, there exists a partition P of [0,2] such that…

Q: 8 Examine holomorphism for given function f(z) = |z|

A: The given problem is to examine the holomorphism of the given complex function. Given function…

Q: Q3/ 1- Let f: Zn → Z10 define by f(x) = 2x+2. Is fhomomorphism or not, explain your answer.

A: Given :-  let f : ℤn → ℤ10 define by f x = 2x + 2.  To Determine :-  Is f homomorphism or not.

Q: Show that f(r) = O(|x – xo|*) and g(x) = o(]x – roP) imply f. g(x) = o(|x – xo]k+i). - -

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Q: Use the Squeeze Theorem to show that f(x) = x² is in R[0, 3]. %3D

A:

Q: Let f(x,y) = x - 2xy – y“. Compute f(3,0) and f(3, – 7).

A: To compute f(3,-7) Given:- f(x,y)=x2-2xy-y2

Q: 6. Consider the function f Cx)=5(o+5)-25 if X#し if xfO D.IS fCx) continuous at X-0? EXPlain

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Q: Give an example where | f(x)dg(x) exist even through f is unbounded on [a, b].

A:

Q: Determine whether k(x) is continuous at r = 3, where x + 3, x 2.

A: The given function is  kx=x+3 , x<0x2+1, 0≤x≤22x,      x>2 A function f(x) is said to be…

Q: (b) Show that the u(x, t) can be written as u(x, t) = 0(x – at) + 4(x + at), where o and are…

A: The multivariable function u(x,t) is a rule that associates or fixes a number to a particular point…

Q: xMyn x2+y2 (х, у) 7 (0, 0) f(x, у) %D (х, у) %3D (0, 0)

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Q: Show that the functionf (x ,y)=xy³onD: x|<2, ]y|<oodoes not satisfies Lipschitz condition.

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Q: Show that x is continuous everywhere. r(x)-x-0 XX<0

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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: Prove that the function f: R -→ R defined by f(x)= x³ is bijective.

A: If a function is both one-one and onto, then the function is bijective.

Q: Derive a finite difference approximation to of(x,y) + f(x,y) using the points f(x, y), ƒ (x+h, y),…

A: Here we use taylor series to find this.

Q: Prove that f (z) =zz is nowhere analytic

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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: Find the Wronskian for the set of functions.{x, ex, sin x, cos x}

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Q: Let f:Z → Z be defined by f (x) = 2x + 1 where x € Z. Is f a one to one correspondence (bijection)?

A: Now let

Q: Q2. Show that F(x) = 3 – x², x € is contractive on its domain, but does not have any fixed points.…

A: To show - Show that Fx = 3 - x2, x ∈ -14, 14 is contractive on its domain, but does not have any…

Q: Let AC C([-1,1]) be defined by A = {f€c'(I-1, 1), f(0) = 0, f'(x)| < 1, V x€ (-1,1)} Prove that A is…

A: The set A is defined as A=f∈C1-1,1, f0=0, f'x≤1, ∀x∈-1,1. To prove that A is relatively compact.…

Q: Define f: R → [2, 0) by f(x) = x² +2,Vx € R.Prove that f is onto.

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Q: 7. Show that the function f : R → R* defined by f(x) is not onto. x2 +1

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Q: Prove that the function f: R W {2} R {5} defined by - 5x + 1 f(x) = #-2 is bijective.

A:

Q: Find aland U so that the function S 7z³ – 3x² + 7, z -2 is both continuous and differentiable. al =

A: To Determine: fins a and b so that the function is continuous and differentiable given: we have a…

Q: The function on ℤ defined by f(x)=-x^2-5 is onto. True or False ?

A: The given function is                  f(x) =x2-5   Graph:

Q: -B-Let f:( M, (Z).+-)-(M, (Z). +..) be a function define by such that g[]=[ EM₂(2). Is f is…

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Q: Prove that the limit of xy[(x^2 - y^2)/(x^2 + y^2)] is zero as (x, y) goes to (0,0)

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Q: 8. If f is conformal on a domain D which is symmetric with respect to the real axis, is f(z)…

A: Given that f is conformal on a domain D which is symmetric with respect to the real axis.

Q: Let f(x) x² and g(x) = xf(x). Find c = (0, 1) as guaranteed by the Cauchy Mean Value Theorem.

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Q: Consider the function h : Z x Z → Z ×Z defined by h(x, у) — (2ху, 7x — Зу). (a) Is h injective? (b)…

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Q: -B- Let f( M,(Z).+.)-(M,(Z),+..) be a function define by such that g( v: al e M,(Z) . Is f is…

A: The given question is epimorphism. Given that,      f : M2ℤ , + , · → M2ℤ , + , · is a function…

Q: O Show that the function f: N2 N defined by f(x, y) = 2X3Y is injective but not surjective.

A:

Q: 5.11. Prove that the function f(z) = Arg(z) is discontinuous at each %3D point on the nonpositive…

A: Solution

Q: 2. Determine whether the following functions are linearly dependent or not by applying the…

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Q: Let f(x) = x² and g(x) = xf(x). Find c = (0,1) as guaranteed by the Cauchy Mean Value Theorem.

A:

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)

Q: Let f be the function from R to R with f (x) = x2. Is f invertible?

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Q: 11. Verify Rolle's theorem for: c) f(x) = In () on [1,3] %3D

A: