Let f be a mapping from [1,+*[ to [1. +[ defined by f(x) = x+1/x. Then f has a unique fixed point O f(x)-f(y)Is2|x-yl and f has no fixed point. O if(x)-f(y)Is2|x-yl and f has a fixed point, O None of the choices.

Q: 5x +1 Prove that the function f: R – {3} –→R – {5} defined by f(x) is a bijection. x- 3

A: Consider the function f:R-3→R-5 defined by fx=5x+1x-3. To prove that f is a bijection, we have to…

Q: Let f be a mapping from ]0.1[ to ]0, 1[ defined by f(x) = x². Then O f has a unique fixed point O…

A:

Q: 4. Suppose that T : R* → R² is a linear map defined by T (x1, x2, x3, x4) = (2x1 x2 + x3, -2x1 – x2…

A:

Q: Let C([0, 1]) denote the set of all continuous functions f : [0, 1] → R, and let Qn (0, 1] =…

A:

Q: Given the map f: R3 R4 defined by f(x,y,z) = (0,0,x,0) then rank of f is: O A. 3- O B. 2- OC.- OD.…

A:

Q: Let ƒ be a mapping from [2, +∞ [to, +∞[defined by: 3 f(x) = x + = x f has a fixed point. O False O…

A:

Q: Prove that the multiplication map · : ℤ/n × ℤ/n → ℤ/n given by [x] · [y] = [xy] is well-defined.

A: A multiplication operation defined on ℤ/nℤ×ℤ/nℤ→ℤ/nℤ as                x·y=xy

Q: Let f(x) = x² – 2 and g(x) = x² + 1. Compute the composition function (f o g)(x). O A. x² – 1 О в.…

A:

Q: Theorem 8.1. The following are equivalent: (1) X is connected. (2) There is no continuous function f…

A:

Q: Please solve this question.

A: We are given that L(p(x)) = p(0). Let p(x), q(x) in P4(F) and α, β in the field F. Now,…

Q: Let f be a function defined on an open interval I containing a. Prove that f'(a) exists and is equal…

A:

Q: Let f be a mapping from [1,+[to [1, +[defined by f(x) = x+1/x. Ther f has a unique fixed point…

A: It is given that ;  f: [1, +∞) → [1, +∞)  defined by  f(x) = x+1x. We have to choose the correct…

Q: Theorem 7.5. A function f : Rstd Rstd is continuous if and only if for every point x in R and ɛ > 0,…

A:

Q: Let f : R² → R² be a continuous function. Prove that f([0, 1] × [0, 1]) is compact.

A: It is given that f:ℝ2→ℝ2 be a continuous function. Prove that f0, 1×0, 1 is a compact set. Let Gi:…

Q: Let X be any set and let Y = "2¬X = {x : X → {0, 1}}, that is, Y is the set of all functions from X…

A: To prove that F is a bijective mapping,we have to show that it is both one-to-one and onto.

Q: Let f be a mapping from ]0,1[ to ]0, 1[ defined by f(x) = x². Then * O If(x)-f(y)|<2|x-y| and f has…

A: A function f(x) is said to have a fixed point if there exists an x in domain such that .

Q: Let g : R → R be a positive function such that g(0) = 1 and g(x+y) = g(x)g(y), ∀x, y ∈ R. Show that…

A:

Q: 2. Consider the function f(z) = ( -3) on (0, 1]. (a) Prove that / for r, ye 0, 1], (b) Prove that /…

A: Contraction mapping theorem: Let (M, d) be a metric space. A mapping f:X→X is contraction mapping,…

Q: Let f be a continuous function on R such that limr→∞ f (x) = L* and lim,-0 f(x) = L-. Assume L+ > L¯…

A:

Q: f (x)dx = f(c)(b – a). a

A: given: the function f is continuous on the closed interval [a, b], then there exists at least one…

Q: 2. Let g : R → R be a bounded function and define f : R\ {0} → R by 1 f(x) = g(x)+ x2 Using only the…

A:

Q: Let f be a mapping from [1,+0[ to [1,+00[, defined by f(x)=x+1/x. Then * f is neither continuous nor…

A: Function is continuous in whole domain I.e [1, inf) by graph.  Hence option A discard.  Since this…

Q: Let g(z) Then there exists an analytic function Ø(z) with Ø(0) # 0 such that: z³ sin(2z) Ø(z) a)…

A: Given function is gz=9z3sin2z. We knw that sinax=eax−e−ax2i Hence,…

Q: True or False A function f has a local maximum at c if there is an open interval I containing c such…

A:

Q: f(c) = g(c).

A:

Q: 5. Determine whether the closed interval [a,b]is homeomorphism to the closed unit…

A: This is a problem from topology.

Q: Given the map f:R3 R4 defined by f(x,y,z) = (0,0,0,0) then rank of f is:

A:

Q: Let a function g be defined on R2 with g(x,y) = (xy(x2-y2)exy) / (x2+y2) Use the following to…

A: Limiting value of g.

Q: Given the map f: R3→R4 defined by f(x,y,z) = (0,0,x,0) then nullity of f is: А. 0- В. 2. С. 3- O D.…

A: Given f is a function from R³ to R⁴ such that f(x,y,z)=(0,0,x,0). Let u=(x,y,z) belongs to Null(f).…

Q: Let f : (0, 1] → (0,1) be the function defined by f ) = „ for all n E N, and f(x) = x if æ E (0, 1]…

A:

Q: Let f be a differentiable real function defined in (a, b). Prove that f is convex if and only if f'…

A: A function f is convex in a domain a,b if for all 0≤λ≤1, fλx1+(1-λ)x2≤λfx1+1-λfx2  ∀x1,x2∈a,b. Also…

Q: True or False A function f has a local maximum at c if there is an open interval I containing c such…

A: We know that, from the definition of local maximum or relative maximum, we say that f has a relative…

Q: Let X be any set and let Y = 27X - {x : X {0, 1}}, that is, Y is the set of all functions from X to…

A:

Q: Let C([0, 1]) denote the set of all continuous functions f : [0, 1] → R, and let Qn (0, 1] = {r1,r2,…

A:

Q: Let ƒ : R → R be a continuous function such that f(0) 0. Prove that f(1)< 1 for all real x.

A: given f:ℝ→ℝ is a continuous function such that f(0)≤1 for all real x f(x)2-3f(x)+2≥0 to prove f(1)≤1…

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: 1-a) Sketch the domain of the function f(x,y)=ln(4x^2-y) (b) Draw a contour map of f(x,y)=x^2+4y^2…

A: Given function f(x,y)=ln4x2-y2

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

A:

Q: Prove that if f is a continuous function with domain [0, 1] and X is the set of all x in [0, 1] such…

A: F is a continuous function with domain (0,1) and X is the set of all x in (0,1) such that F(x) = 0.…

Q: Let f: R/{1} → R given by f(x) ==, for every x € R/{1}. Then the given f is %3D x-1 bijective.

A: Topic:- function

Q: Let f be a mapping from ]0.2[ to [1,+ [ defined by f(x) = 1/x. Then * %3D None of the choices f is…

A:

Q: Suppose that f1 : [0, 1] → R and f2 : [0, 1] → R are continuous everywhere and that f1(0) f2(1).…

A: Intermediate value theorem : Suppose be a continuous function then it takes all values between .

Q: Let f:R" → R be a continuous function, and let x1, x2 € R" be distinct. Prove that there exist a…

A: Given: Now consider the given, Let f:ℝn→ℝ be a continuous function, and x1, x2 ∈ℝn  be two…

Q: Let f be a mapping from ]0,2[ to [1,+[ defined by f(x) = 1/x. Then * f is a homeomorphism None of…

A: The given data are: Let f be a mapping from ]0,2[ to [+1,∞[ by using the given function fx=1x.

Q: Let f be a mapping from ]0,1[ to ]0, 1[ defined by f(x) = x². Then * O f has a unique fixed point O…

A:

Q: Let g : Zx Z-ZxZ be the function given by g(x,y) = (3x + Y, x – y). Find, with proof, an element of…

A: Take a point (1, 0) and suppose it belongs to the range set.

Q: Prove that if f is a continuous function with domain 0, 1] and X is the set of all x in 0, 1] such…

A: Given that f is a continuous function with domain 0,1. Let, X=x| x∈0,1 and fx=0. Also, given that s…

Q: Given that f is an invertible function, if the point (x, y) lies on the graph of y = f(x), then the…

A: We have to find at which point is guaranteed to lies on the graph of  y=f-1(x).

Q: Prove that f(x) = x 3 - l2x is differentiable. Compute f'(x) and find f '(-3), f '(0), f' (2), and…

A: Given information: The function is f(x)=x3-x. Calculation: We need to find the value of f'(x) and…