Show that if f : A → B and g : B → C are one-one, then gof : A → C is also one-one.

Q: Prove that for all a € R, 0 < r² + 2 VI

A: Prove the following

Q: Prove that 1/r + 1/r2 + 1/r3 + . . . = 1/r − 1, for |r|  &gt; 1.

A: we have to prove that: 1r+1r2+1r3+. . . . . .=1r-1  ,  for r&gt;1

Q: Let G= Z* and let * be an operation in a * b. Is a * b = ab/2 is closed under the operation?

A: Disproof by an example.

Q: If ab is algebraic over F and b ≠ 0, prove that a is algebraic over F(b).

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Q: Answer with True or False. 1. The functions from Z to Z is onto if f(n)=n−1. 2. These function…

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Q: For a, b, c, and d: Let h(x) = (f og)(x) a) Let h(x) = (2x-5) and f(x)= x, find g(x). b) Let h(x) =…

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Q: Evaluate the sum of with x1 = 0, x2= 2, x3 = 4, x4 = 6, and Ax= 1, for f(x) = 5x -2. i=1 4. i=1

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Q: Let g(x) -x V25- x. A+ where (25 -je" A 9"(x) where (25 - x)"

A: As per our guidelines, we are allowed to answer only one question. Since, you have asked multiple…

Q: Suppose f(x) : a.x+6 for real numbers a, b, c with both a and c not zero. сх—а (a) Find the domain…

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Q: If f(1) > 0 and f9) < 0, then there exists a number c between 1 and 9 such that Ac) = 0. く

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Q: Let E be an extension of F and let a and b belong to E. Prove thatF(a, b) = F(a)(b) = F(b)(a).

A: Given Data E be the extension of F, and a &amp; b belong E.   The E is an extension of F, and…

Q: Suppose that f(x) = x², xER, and g(x) = 25 + x, xER. (a) Show that (fo g)(X) = (25 + x)², xeR. (b)…

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Q: Show that for all x, e* > 1 +x.

A: It is given that, ex≥1+xwe have to show that for all x.

Q: Show that for all x>0, e* >1+x+÷ 2.

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Q: Prove that if f : [-1,1] → R is measurable, then |f| is also measurable.

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Q: Let f(x) = 2x +3, g(x)= x- 4, and h(x) = 4x- 1. Compute (fo ho g)(2).

A: Here we have to find the composite function.

Q: Prove: If f1, f2 ∈ R(a) on [a, b] with f1 ≤ f2, then ∫ab f1 da≤ ∫ab f2 da

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Q: Let f lx):3x,9J=xland hlx): Ne Express each function below as a Composition of any twO of the above…

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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: Suppose that .3 f(3) = }, and f(x) > 0, and juw <0. < 0. Is it possible that f(2) = . %3D False O…

A: You have asked two different questions. As per the company guidelines we are supposed to solve first…

Q: Let =[2,0}, then f is o- algebra.

A: Given a statement

Q: • The domain of f(x) 3-x is 1-x2 %3D 1. R 2. R/{1,-1} 3. R/{3} 4. (0, 0) 1 0 2 3 O 4 O

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Q: Let * be the operation on the cartersian product of real numbers R *R such that (a,x) * (b,y) =…

A: Given that, let * be the operation on the cartesian product of real numbers R*R  such that (a, x) *…

Q: For f(x) = x - 4x + 3, there exists a'c' in [1, 3], which satisfies the conditions of Rolle's…

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Q: Which of the following are proper interpretations on f(2)?

A: Proper interpretation on f’(2) f’(2) is the derivative of function f at x=2 Derivative is the slope…

Q: Suppose f: A → B, f is one-to-one and onto, and Y c B. Show that the inverse image of Y under f and…

A: The function f is one-one and onto.   The function f is one-one if two different element in A…

Q: If F(x)=2x^2+5x+2,then F'(x)=4x+5.Thus,if f(x)=4x+5,then f is the___of F and thus F is a/an___of f.

A: F(x) = 2x^2 + 5x + 2 F'(x) = 4x + 5

Q: Prove that f (x) = -16+x 4 and g(x) = = 4x + 16 are inverses. ||

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Q: Given that f(x)=2x+1, and g(x)=x²-x+3. Determine (gof)(x), expand, and simplify.

A: From the functions..

Q: Let f(x) = 2x +3, g(x) = x – 4, and h(x)= 4x – 1. Compute (h o go f)(3).

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Q: that f(−4)=5, f'(−4)=−2, g(−4)=3,

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Q: Let f be the function from {a, b, c, d} to {1, 2, 3, 4} with f(a) = 4, f(b) = 2, f(c) = 1, and f(d)…

A: For bijection, the function needs to be one to one and onto.

Q: Prove that if f(z) is analytic, then: [f'(2)² = (² |f' (2) 1³² = (²+ (²)² = (²+)²

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Q: Evaluate the sum of 2 f(x;) Ax, with x, = 0, x, = 2, X3 = 4, x4 = 6, and Ax = 1, for f(x) = 3x – 7.…

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Q: 3. If f:(X,r)(Y.A) and g:(Y,A)→(Z,r) are open functions, prove that gof :(x.r)(Z.r) is open…

A: Definition: An open map is a function between two topological spaces that maps open sets to open…

Q: a field and let c,d ∈ F. Show that c⋅(−d) = −(c⋅d).

A: Associative Property of Field F for a,b,c∈F a·b·c=a·b·c

Q: Let f(x) = 4x + 9, x, =2, x2 = 3, x3 = 4, X4 = 5, and Ax = 1. %3D (a) Find Ax. i= 1

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Q: Show that the mapping f:R2 → R² defined by f(x1, x2) 5x2) is linear. = (2x1 + x2, -3x1 +

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Q: Consider f : {1, 2, 3} → {a, b, c} given by f(1) = a, f(2) = b and f(3) = c. Findf-1 and show that…

A: Given function is

Q: The Maclaurin serles for f(x) = x²elz) is: (-2)*z*+2 (b) 00 (e) None (2)-kyk+2 2(k)xk+2 (c) 2kyk+3…

A: Any function which is differential then the function can be written in the polynomial  A Taylor…

Q: Q2) If f(x) = (x), and g(x) = (2- x)i find (gof) (x) and the domain of (gof)(x).

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Q: Let x ∈ R. Prove that if x 0.

A: Given :

Q: Let E be an extension of F, and let a,b be elements of F. Prove that F(a,b) = F(a)(b)=F(b)(a)

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Q: Define f, g: R → R as if f(x) ={3x – 2 if -13 and g(x) = 8x – 1. Find f o g and gof. %3D

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Q: 2. Let f(x) = ax+b cx+d (a) Show that if f is one-to-one, then ad bc + 0. (b) Show that if ad – bc +…

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Q: Show that f(x) = x* + 2x? + 5x +1 is irreducible over Q.

A: The given polynomial of the function is f(x)=x4+2x2+5x+1 We have to show that the polynomial is…

Q: Prove that F (3x)(A → B) → ()A → (3x)B.

A: We have to prove, ∃xA→B→∀xA→∃xB

Q: Suppose that f(x) = x" + ax"-1 + . . . + a, E Z[x]. If r is ra- tional and x - r divides f(x), show…

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Q: If f is invertible and f(4) = 14, then f(14) = and (f(4)) - %3D

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Q: recall the definition of |A|=|B| for sets A and B and use that to show  |N|=|{ n/2 : n∈ Z}|  no…

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