Let f, g,h: A A be any bijective functions. Then (f ogoh)-1 h-1 og-1of-1 Ois equal to the above None of these f-1oh-1og-1 O is equal ta the above is equal to the above

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: Assume that a b•c and that gcf(a,b) = 1. Prove that a lc (hint: use the result that gcf(a,b) = 1 iff…

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Q: 2. Prove the general formulas (GF)» = G«F* and (F-')» = (F«)- in the special case where F and G are…

A: Given G and F re isometries in ℝ3 To prove  1.GF*=G*F*2. F-1*=F*-1 Since F and G are isometries in…

Q: Use direct prove to prove that for any two positive numbers x and y, x+y Vxy s 2

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Q: Prove that the quartic function q(z) = r* - 16 has only

A: Given that: q(x)=x4-16

Q: Find f, and f,, and evaluate each at the given poi 2xy f(x, y) = (1, 1) 2x + 2y2 Fx(x, y) %3D f(1,…

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Q: Prove that f : R – {3} → R – {1}, defined by f(x) = is a bijection. x-1 x-3

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Q: Let f(T) and g(T) be polynomials in Q[T] whose gcd in Q[T] is 1. Show that 1 is also their gcd in…

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Q: Q: If f:R R is defined by f (x)=2x+7 Prove that f is a bijecti on Also find the inverse of f.

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Q: Prove that if a e R, then a 0 = 0 = 0 · a. Suppose x, y, z E R. Show that xz = yz if and only if…

A: We will prove the given statements.

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

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Q: Let f, g,h:A + A be any bijective functions. Then (f ogoh)-1 f-1 oh-1 o g=1 is equal to the above…

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Q: Let f : A → B and g : B → C be bijections. Prove that gof is invertible and that (g o f)-1 = f-1 o…

A: Problem based on function.

Q: Define g : (-1, 1) → R by g(x) 1- x2 " Show that |(-1, 1)| = |R| by showing that g is a bijection

A: Concept Used: For a function f:A→B If f is a bijection, then |A|=|B| Consider the function g:(-1,…

Q: Find the supremums of the following sets [1 :by proving your results .Si = {a < x < b; x E R} (a %3D…

A: Since, the question is of multiparts, according to the Bartleby Answering rule, only 1st questions…

Q: 1 1 b. From (i) prove that, xb + x ² + 1 + x² + x ² + 1 ² + 1 = 1 xa +x+b+1 c. From (ii) given in…

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Q: Show that the Jacobian of a function f on E3 is given by the triple product 7.46. af af af J(f) 11…

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Q: Let H(x):=k for x=, (keN) and H(x):=0 elsewhere on [0,1] Show that H(x) 4ER[0,1].

A: In this question, we use the definition of Riemann integral to check the function H(x) is Riemann…

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]…

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Q: Let f,g,h: A→ A be any bijective functions. Then (fogoh)-1 f-1og1oh-1 h-1og-1of-1 O is equal to the…

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Q: Which of the following is indeterminate at x = 1? x² + 1 x2 – 1 x2 – 1 Vx +3 - 2 x2 +1 Vx + 3 – 2 x…

A: We plug x=1 in each expression

Q: Suppose that the functlons u and w are defined as follows. u (x) =x² +3 w (x) = /x+ 2 Find the…

A:

Q: 2. Let k be a fixed real scalar. (a) Show that L: R? R defined by a ka +b

A: Given, k be a fixed real scalar. (a) To prove that Lk: ℝ2→ℝ2 defined by…

Q: Which of the following is indeterminate at x = 1? x2 +1 х2 — 1 x² – 1 x² +1 х — 1" x +2 Vx +3 – 2'…

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Q: Consider the following bijections on R: f (x) = - g(x) = x – 7 Then (f o g)¯'(x) =

A: Given, fx=-x,gx=x-7 Find the value of f∘g-1x.

Q: Let f : A → B and g : B → C be bijections. (a) Prove that (go f)-1 exists. (b) Prove that (g o f)-1…

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Q: 2-Express each function in the form u(x, y) + iv(x, y), where u and v are real, (c) ez2 (d) In(1+…

A: We have to write in u(x,y) and v(x,y) form.

Q: To get a feel for higher order ODES, show that the given functions are solutions and form a basis on…

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

A: Consider the given information: The given function is, f:ℝ-{2}→ℝ-{5} f(x)=5x+1x-2 To show that f is…

Q: Prove by direct method: if x, y>0  such that x ≤ y then √x≤√y

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Q: 90. Let z = f(u, v), u = ax + by, and v = ax – by. Express z and in terms of fu,fy, and the…

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Q: Find the source Q cx) Such that the So lutiim Uckit) of + Q(x) 2 ə x² 0 < x <∞ | U(x,o) = 0

A: given equation ∂u∂t=k∂2u∂x2+Qx ,       -∞&lt;x&lt;∞            ...iux,0=0 claim- find the source Qx…

Q: 5x+1 14. Prove that the function f: R \ {2} → R \ {5} defined by f(x) = is bijective. x-2

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Q: Let a, y, z be positive real numbers such that r +y+z xyz. Prove that 1 1 1 1 1 1 1+ x2 1+ y2 1+ 22…

A: Given- x + y + z = xyz This inequality can be proved by magic no. Concept-  First we will prove…

Q: Suppose m, n ∈ Z + are relatively prime. The function f : Zmn → Zm × Zn defined by  f([Z]mn) =…

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Q: Prove: let v: X Y bijective then V inverse(v(x)) = X

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Q: Let R→R be defined by g(x) 2y+1 2y+2 Is g bijective? Prove or give a counter example.

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Q: Prove that for every y E R, there exists a unique x E R such that 2(x – 1)3 – y = 0. %3D

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Q: 10.55. Let A = R – {1} and define f : A → A by f (æ) =, for all æ E A. a. Prove that f is bijective.…

A: A=R-1f:A→Afx=xx-1

Q: Let ƒ : [0, 1] → R be given by f(1) = xr and g : R → [0, 1] be given by g(x) = Prove that |[0, 1]| =…

A: Cantor-Schrӧder- Bernstein Theorem: if there exist injective functions f: A → B and g: B → A between…

Q: (a) Evaluate the integral [21,22,23,21] 3(z − 1)²dz, z₁ = 1, %2 = 1 + i, z3 = −1 + i. (b) Show that…

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Q: Determine the largest subset S of C where the function ƒ(z)=3x² + 2x−3y² −1+i(6xy+2y) is analytic;…

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Q: Let ƒ : [0, 1] → R be given by f(x) = x and g : R → [0, 1] be given by g(x) = Prove that |[0, 1]| =…

A: Cantor-Schrӧder- Bernstein Theorem: if there exist injective functions f: A → B and g: B → A between…

Q: Define g : Z → Z × Z by if t is even if t is odd S (t, t + 3) g(t) = I (t, 2) Is g injective? Is g…

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Q: Show that The Hermite polynomials can be difined by the integral representation H, (x) = (-i)"2"e**…

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Q: . Let f € C[a,b] and g E R[a, b] with g > 0. Prove that 3c E [a, b] such that (2)g(x)dzx = [(e) […

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Q: 6. Find an explicit function f: [0,1] and prove why it is (0, 1) that is a → bijechove bijection

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.