Find functions f and g from the set of positive integers to the set of real numbers such that f (n) is not O(g(n)) and g(n) is not O( f (n)).

Q: Find k such that f(x) = x3 - kx2 + kx + 2 has the factor x - 2.

A: The general polynomial equation of degree n  is an equation of the form…

Q: 1. Explain why both f(x)= x³+ 1 and g(x) tions. x3+1 are rational func-

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Q: Let f(x) = x3 + x² + 4x – 1 € Z5[x]. Then x = -1 is a root for f(x) of multiplicity: 4 O None 3

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Q: If f1(x) and f2(x) are functions from the set of positive integers to the set of positive real…

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Q: Find the constant a such that the function is continuous on the entire real line. 7sin x if x < 0…

A: Here we find the value of constant a

Q: Find the value of c such that the function is continuous on the entire real number line. f(x) =    x…

A: Given that the function: fx=x+3,              x≤2cx+6,           x&gt;2

Q: Show that the function f: R - {2} (arrow) f: R - {5} defined by f(x) = (5x+1)/(x-2) is a bijection.

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Q: Find (a) the domain of  f and (b) f-1 and its domain.   f(x)  = In  (2 + In x)

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Q: Show that the function g: R - R defined by g(x) = 4x² + 3 is neither one-to-one nor onto

A: To show the function g:R→R defined by gx=4x2+3 is neither one to one nor onto. Solution: First we…

Q: Let ƒ(x) = x3 - x - 1. Use the Intermediate Value Theorem to show that ƒ has a zero between -1 and…

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Q: 2. Find the constants a and b such that the function is continuous on the entire real line. 7 x 4

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Q: The domain of f/g consists of numbers x for which g(x) __________  0 that are in the domains of both…

A: we have to fill the following blanks ;  The domain of f/g consists of numbers x for which g(x)…

Q: Let f (x) = x&gt;(x2 - 1), g(x) = (1 - x)&gt;(1 + x), and h(x) = 2&gt;(3x + 1). Express as rational…

A: fx=xx2-1,  gx=1-x1+x  &amp;  hx=23x+1 Now, gx+1=1-x+11+x+1=1-x-1x+2=-xx+2

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: Show that the function g: R -R defined by g(x) = 4x2 +3 is neither one-to-one nor onto

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Q: 12 f(x) = X ¹²³ + x² + x² + x² Show that whether f(x) is irreducible or not

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Q: Compute f '(a) algebraically for a = 3. f(x) = 2x^2+ x

A: Solution: The objective is to find f '(a) algebraically for a = 3

Q: Show that there are no polynomials f(x) and g(x) in Z[x] such that 1 = 2f(x) + xg(x)

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Q: Prove that the function f(x)=x^2 defined on the domain of all real numbers does not satisfy a…

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Q: Let k(x) f(x)g(x) h(x) If f (x) = 3x, g(x) = x - 2, and h(x) = -x² + x + 1, what is k' (x)? %3D…

A: We know that, from quotient rule of differentiation We have,  fx=3x, gx=x-2 and hx=-x2+x+1 Now,…

Q: Prove f:N - N. F(X) = 5x is injoctive.

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Q: Find the constant a such that the function is continuous on the entire real line. 2sin x a 11 g(x):…

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Q: Let f(x) and g(x) be cubic polynomials with integer coefficientssuch that f (a) = g(a) for four…

A: Consider

Q: 15.B. Show that a polynomial function f, given by f(x) anx" + an-1x"¬1 +... + a1x + ao, x€ R, is…

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Q: 3. Let P(x) be a polynomial of degree n. If n is even, prove P(x) cannot have an inverse function.

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Q: If f^(x)=c for some natural number n and constant c, then f is a polynomial function. True or false?

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Q: Find the domain of the function. h(x) = In x + In(2 - x)

A: Given h(x)=ln x+ln (2-x) Find the domain by finding where the expression is defined.

Q: 3. Use the Intermediate Value Theorem to show that f(a) = a2-3x +1 has at least one real zero…

A: According to intermediate value theorem, if f(x) is a continuous function whose domains contains the…

Q: 6. (a) Express f (x), where x3 + 1 f(x) = %3D (x – 1)2 (x² + 1) Hence, or, otherwise, find HE AND…

A: f(x)=x3+1(x-1)2(x2+1)

Q: Show that f (x) = 3x² - x - 1 is O(x²) with suitable C and k.

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Q: Prove that the function f(x) = x3 – 6x2 + 15x - 18 is strictly increasing on R. %3D

A: Given function f(x) = x3 - 6x2 +15x -18 We need to differentiate the given equation  f'(x)= 3x2 -12x…

Q: 3. For f (x) = 2x³ + 4x² + 3x – 7, prove that there exist a number k such that f(k) = 20.

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Q: The polynomial function f ( x ) = x 4 − 19 x 3 + 134 x 2 − 414 x + 468 has 5 + i as a zero. Select…

A: Given that f(x)=x4-19x3+134x2-414x+468 has one zero at x=5+i. We need to find the other 3 zeros.

Q: 1. Use the Intermediate Value Theorem to show f(x)= 2x +6x² – 3 has a zero between a = -3 and b =…

A: I am attaching image so that you understand each and every step.

Q: Find the domain of the function. k(x) = In |x|

A: The domain of a function, f(x)  is the set of input values of x for which f(x) is real and defined.

Q: Find a polynomial f(x) of degree 4 with leading coefficient 1 such that both  −7 and 2  are zeros…

A: Since the zeros are multiplicity of 2. Zeros = -7, 2 The value of leading coefficient is 1 given in…

Q: The domain of the function f(x) = |x +5| – 2 is: O (-∞0, 3) U (3, 0) O 3, 0) All real numbers O (…

A: The objective of the question is find the domain of the given function.

Q: Find a third-degree polynomial function f(x) with real coefficients that has -3 and i as zeros and…

A: Polynomial: The polynomial which has real co-efficient and of third degree. f(x) has three zeros…

Q: Compute f' (a) algebraically for a = -4. f (x) = -x² – 21 O f' (-4) = 8 O f' (-4) = 24 O f' (-4) =…

A: Derivate f(x) with respect x and then put x=-4

Q: Find the formula of f(x) a polynomial function of a least degree

A: At x=-4 the curve cuts the x-axis. So x=-4 has multiplicity=1 So the factor is (x-(-4))=(x+4)   At…

Q: find an integer n such that f(x)= 0 for some x between n and n+1. f(x) = x^3 - x + 3

A: Given:The function f(x) = x3−x+3.Known fact:Newton Rphson's method.It is an iterative process to…

Q: Let f be an entire function such that |f(2)| < A + B|z|" for all z E C, where A and B are positive…

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Q: Let f(x) = 2x - 5 g(x) = 4x – 1 h(x) = x² + x + 2 Evaluate. (g of)(0) O-21 O -7

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Q: Prove that for a > 0, the function

A: Introduction: The given function is:

Q: Let f(x) be a function on (- 0, 0) and f(x + 2) = f(x - 2). If f(x) = 0 has only three real roots in…

A: Introduction :

Q: 12. Let f(x) = x + 2 and g(x) = x² - 1. Find (gᵒf)(4)

A: Given two functions 1.f(x)=x+2 g(x)=x^2-1 lets find g(f(x)) at 4

Q: Find (f 1)(a) for f(7)=0, f'(1)=-1, and a=0. -л -1 1/л -1/n

A: topic - derivative of inverse function

Q: The domain of function x) = In(2– x) is %3D

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Q: 3x + 7x2-5x-4 Let f(x) = %3D 6x +4 1. Use Ad-hoc calculations to prove that f(x) is O(x2) 2. Use…

A: let fx=3x3+7x2-5x-46x+4 1) use the Ad-hoc calculations to prove that fx is Ox2 2) use limits to…

Q: 5. Find the domain of the function: y = In () x-4

A: y=lnx-1x-4