Let f : R → R be defined as f(x) = 3x. Choose the correct answer.(A) f is one-one onto                              (B) f is many-one onto(C) f is one-one but not onto               (D) f is neither one-one nor onto

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.

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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

A:

Q: Let f : R R with f(x) = -2x +5 Prove that f is onto.

A:

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A:

Q: Determine if the function f : RxR → R, defined by f(x, y) = x2", for all (x, y) E R×R, is…

A: f is not one-one  But, f is onto.

Q: Prove that f: R R where f (x) = |x| is neither injective surjective. nor

A: To prove this theorem, we use the defination of one one and onto.

Q: Let f : (0, ∞) → R be defined by f(x) = ln x. Prove that f is onto.

A:

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

A:

Q: Show that the function f : R → R,defined as f(x) = x2 , is neither one-one nor onto.

A: We will show this by counter examples.Let x= 2 and -2 Then f(2)= 2^2 = 4 and f(-2)= (-2)^2 = 4 This…

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.

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A: Given data is: Let f be differentiable on R. Suppose that f'x≠0 for every x. To prove f has at most…

Q: Prove that the function f(x) = 2x is uniformly continuous on R.

A:

Q: 2. Show that if f : R → R is continuous on R and f(r) = 0 for all r E Q then f(r) = 0 for all x E R.

A:

Q: Prove that the function g: R &gt; R defined by f(x) = 1+2^x is one-to-one but not onto.

A: Consider the function f:R-&gt;R defined by:

Q: Let f : Z → Z be such that f (x) = x + 1. Is f invertible, and if it is, what is its inverse?

A:

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

A:

Q: Exercise 3 Define f : Rª\{0} by f(r) = |r|,'. Using the e-5-definition, prove that f is continuous.…

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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: f is continuous at a if and only if |f| is continuous at a.

A: False

Q: Let f(x, y) = xy - 4x on

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: Prove that f (x) = x-12 is : (i) not sectionally continuous in every finite interval in the

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

A:

Q: Let f be a function defined on R satisfying If(z) - f(y)| < 히 - J), V,yER. a. Show that f is a…

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Q: Let f(z) := z5 . Show that f is invertible in a neighbourhood of i and calculate (f-1)'(f(i)).

A:

Q: A function from ZxZ to Z defined by f(m, n) = m-n is an onto function. True False

A: Here we will examine that f is onto or not.

Q: Show that the function f : R→ R : f(x) = x° is one-one and onto.

A:

Q: Prove f(x) = x2+4 is not uniformly continuous on R.

A:

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

A: f(x) is continuous, if LHL=RHL=f(x&gt;0), where LHL is left hand limit while RHL is right hand…

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

A:

Q: Suppose that fn ⇒ f on R and that each fn is a bounded function. Prove that f is bounded.

A: Solution

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

A:

Q: Let f: X→Y be a function. Please prove that if f(X-A)=Y-f(A) for any A⊆X, then f is onto.

A:

Q: Show that the function f(x) = x’ + 1 is invertible from R to R.

A: First prove that the  function is one -one  let f(x)=fyx7+1=y7+1x7=y7x=y therefore, the function is…

Q: Let F(x)=∫(from −4 to x) −2cos(t^2)dt. Use the Fundamental Theorem of Calculus to find F′(x).

A: Solution:

Q: 2. Let f: R → R_be f(x) = 1+ |r|" (a) Is f everywhere defined? If not, give the domain. (b) Is f…

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Q: Let A := (0, 1] and let ƒ : A → R be defined by f(x) = !. Prove that ƒ is continuous on A.

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(x) = Va and g(x) = x² – 1. Find go f and determine its domain. |3D %3D

A:

Q: Let A := (0,1] and let f : A → R be defined by f(x) = . Prove that f is continuous on A.

A: Consider the provided question,

Q: Define the function f: R→ R by f(x) = x5 – 1, Vx € R. a) Prove that f is onto? b) Prove that f is…

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

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Q: B. Show that the space V = {f : R → R} of functions from R to R is not finite- dimensional.

A: Let V=f:ℝ→ℝ. Let A is set of all polynomial with integer coefficient. We know that every polynomial…

Q: Consider the function f: Z x Z → Z defined by f (m, n) = 2m – 2n for all (m, n) E Z × Z. (a) Is f…

A: A function f: A→B is said to be one-one if        f(x1)=f(x2)   ∀x1, x2∈A         ⇒x1=x2 In words,…

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

A: