5. (a) Consider the function f : C → R, f(2) = (2 + 2)?. Write down the domain, codomain and range of f. Show that f is not injective. (b) Consider the functions g : R →R defined by g(x) = r² + 4, rER and h: R → R defined by h(x) = 1 – 2, r € R. Is h o g surjective? Is g o h surjective? You may answer simply "Yes" or "No".

Q: f,(x)= e* f2(x)=e¯* (x) = e'

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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: The graph of f(x)= - x2-1 is concave up on interval (-1,1)^O (0,1) 0 (-1,0) U (1, 0 ).co (-∞, – 1)U…

A: Given, fx = xx2-1  We have to find the interval in which graph of  fx = xx2-1 is concave up.

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: 2. Consider the function / with domain (-2, o0): 3|2 – 1| 2 + 27 - 4 if – 2 <z<1 f(z) = if r = 1…

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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: 5. Show that the function h(r, y) defined by if + y 1 is discontinuous.

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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: Let g=yf(xy). Show that: - дд у ду дд - Х = g дх -

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Q: 6. Use definition of continuity to determine the value(s) of k such that the 2х + 5, х< 3 function f…

A: Topic : Continuity

Q: Let f(x)=x² -x² +x + 1 and g(x)= 3-х max{f(t):0sx; for 0 s x s 1} for 1 < x s 2 Discuss the…

A: Topic:- Functions

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: Determine the types of singularities of the function f(z)=1-e^2z/z^4

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Q: 4. Let S = {x EN|3z ENA z? = x)). Let f : N → Sbe the function f(x) = x². Is f 1-1? Onto? Show why.

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Q: Is there a continuous function such that - f: [1,2] U [3,4] → R - f(1) * f(4) < 0 - f(x) not egual 0…

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Q: 2. Show that the function, defined by f(x) = is NOT uniformly continuous on its x2 %3D domain D(f) =…

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Q: Let f, g:R R defined by if x 1 (x2 and g(x) = 3x + 2. %3D Find f-1 and g-1.

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Q: 1. Find of ахду OF f(x,y) = xy (x+y-1) -x2

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Q: 1 Let f (x) = ; 1 , then f (x) satisfies the - - 2 conclusion of Rolle's Theorem on 0, 1| 6.

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Q: Suppose that f(x, y) is a differentiable function of two variables defined on the entire xy–plane,…

A: Let us consider a function f:ℝ2→ℝ defined by  fx,y=x2y2                                      (i)…

Q: Let f(x) = x -[r] and f :[1,5) →[0,1). What is the discontinuously type of f(x) at the point x = 3?

A:

Q: Differential Equestion

A: You have asked a question with multiple sub parts. I will address the first three of them. Please…

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

A:

Q: 5. (a) Consider the function f : C → R, f(2) = (z + z)². Write down the %3D domain, codomain and…

A: The domain is C, the set of all complex numbers. The codomain is R, the set of all real numbers.

Q: If f is a function defined from R to R, is given by f(x) = 5x - 3 then f -(x) is given by %3D -- 20…

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Q: 4. Let f be defined for all real x, and suppose that \fx) – f(v)| < (x – y)? for all x, y e R. Prove…

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Q: 9. Consider the function f: R → R, defined by (x²sin(1/x) ifx # 0 f(x) = if x = 0 Then a. f is not…

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Q: 8 Examine holomorphism of given function  f(z) = |z|2

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Q: Let f (x, y) = 4-y² In(y–z²) %3D Draw a picture of the domain of this function.

A: x,y ∈R2

Q: A critical point xo, Yo of a function f is obtained when O Vf(x0, Yo) = 0 or Vf(x0, Yo) does not…

A:

Q: 4. If continuous function f:[0,1] U (2,3) → R is one-to-one, is f also continuous? 1

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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: 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: The graph of f(x)= x2 -1 is concave up on interval (-1,1)^O (0,1) 0 (-1,0) U (1, 0 )co…

A: Given,  The function fx = xx2-1  We have to find the interval in which the graph of  fx = xx2-1 is…

Q: 1 The domain of the function f(x,y) is the set of all ordered pairs (x , y) such that : Oy + +x Oy -…

A: We have a given function F(x,y) .and finding the domain of function

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

A:

Q: Show (in terms of  −δ) that a function f : [2,7] −→R be defined by f(x) = √x2+1 is uniformly…

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Q: 7.14. Show that if f = u+iv is analytic in a region S and u is a constant function (i.e.,…

A: Analytic function which is a constant

Q: 13. Let g be defined by V7x +2 V6x + 4 | g(x) = if x 2 - and x 2 x – 2 k , if x = 2 For what value…

A: The given functiong(x)=7x+2-6x+4x-2,if x≥-27 and x≠2k,if x=2

Q: Show that f (a) = is Libschitz function on r+ 2

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Q: 1. Show (in terms of -5) that a function f: [2, 7] R be defined by f(r) V+1 is uniformly continuous.

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Q: Compute the linearisation l(x) of f(x) = √1+x3 at x=2 and use this to estimate f(2.05)

A: Given function is: fx=1+x3 At x0=2 So the linear approximation is: Lx≈fx0+f'x0x-x0…

Q: 2.23 Let 1 - x2 f(x): if x € Q, if r ¢ Q. Find all points p at which f is continuous. Prove your…

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Q: Suppose f is a differentiable function on an interval II and f(x) 6= 0 for every x in the interval.…

A:

Q: 3. Show that there does not exist a non-constant continuous function f : R2 → R such that f(x, y) =…

A: Given:- f: R2→R

Q: Σ 1 is continuous function in R. Prove f(x) = 4 n=1 n' +x*

A: The given function is                                     f(x)=∑n=1∞1n4+x4 To show that the above…

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: 9. Show that if the analytic function f(z) has a zero of order N at zo, then f(z) satisfying g'(zo)…

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Q: 5) Suppose a path C is defined by r(t), 2 and r(8) = . If f is a differentiable function satisfying…

A: