(a) Explain with cases why, if a is a real number,  then a ≤ |a|. (b) Explain, if, for any real number a, sgn(a) = 1 if a > 0, sgn(a) = −1 if a < 0 and sgn(a) = 0 if a = 0, then a = sgn(a)|a| and |a| = sgn(a)a.

Q: Find two positive real numbers x and y such that they add to 120 and x2y is as large as possible.

A: Oue Aim is to find two positive real nnumbe x and y such that theyadd to 120 and x2y is as large as…

Q: (2) If h is defined by h(x) = |x| +2, (a) What is the range of h if the domain of h is the interval…

A:

Q: Prove or disprove the statement: For all x E R, |x + 1| + |x – 1| > 2.

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Q: If f(x) = 10x^2-37x+11, then there are two real numbers a<b such that f(a)=f(b)=4, and there is a…

A: The objective is to solve .

Q: (2) The floor function is defined as a| = max{m e Z|m < x}. Effectively, the floor function is the…

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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: The domain of h(x) 7 - consists of all real numbers except 0 and 3, represented in interval notation…

A: Given function is hx=1x+7x-3. To find domain. Solution: Domain of a function is the value of x for…

Q: b) Suppose h is continuous on R and h(q) = 0 for every rational number q. Does it follow that h(x) =…

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Q: Given the function g(x) = In(x? + k?) with k a positive real number, hence k > 0. %3D and g"(x) =…

A: *Graph is concave up on interval (a ,b) if double derivative of function is f" (x) &gt; 0 on…

Q: R is the amount of rain, in inches, on any particular day in 2012, and it is said to be uniform…

A: Wr want to find the probability by chebycheves inequality.

Q: ·C) - G) + G)-G) + +-1 - Prove that for any positive integer n, 0.

A: Use simple property of combination.

Q: In general, if f(x) is real and continuous in the interval from x to xy and f(x1) and f(xu). between…

A: Since you have asked multiple question, as per company guidelines we will solve the first question…

Q: Given (a) fo g(x)= (b) go f(x)= (c) go g(x)= (d) fo f(x)= Note: If needed enter as INF and - as…

A:

Q: Consider the set S = {r E Q : r² < 2}. In the land of rationale numbers is there a least upper bound…

A:

Q: (a) Express in ascending powers of x as far as the term in x' (2+x)' (b) Express in ascending powers…

A: To expand and simplify the given terms in the ascending powers.   Expression In the power of:…

Q: 1. The set of all pairs of real numbers of the form (x, 0) with the standard operations on R2.

A: 4. Given that the set of all pairs of real numbers of the form (x, 0) with the standard operation on…

Q: @[V2]is Principal Ideal Domain. O True O False

A:

Q: The sum of two positive real numbers x and y is 3. Find an x and y among these such that the product…

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Q: Choose the best conclusion to the statement: If f(3) = -1, then lim f(x). x-3 O A. exists and is…

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Q: The sum of two positive real numbers x and y is 12. Find an x and y among these such that the…

A: The sum of two positive real numbers x and y is 12.

Q: Theorem: For every pair of real numbers, x and y, such that x ≠ 0, there exists a real number z such…

A:

Q: Prove that (z - x)(z - y) = (y - x)?. a:= is not a rational number. Hint: Express (x - x) and (z -…

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Q: (2) If h is defined by h(x) = [x| + 2, (a) What is the range of h if the domain of h is the interval…

A: (a) , (b).

Q: . Suppose f (x) is differentiable for any real value x andf'(x) 2 such that f(c) = c². (Hint: use…

A:

Q: Z is the set of integers, R is the set of real numbers. 1) Functions f, g, h are defined as f: R ->…

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Q: nd the supremum and infimum of the set D defined by D:= :neN

A:

Q: (2) If h is defined by h(x) = |x| +2, (a) What is the range of h if the domain of h is the interval…

A: Given function is hx=x+2 Graph of the function can be drawn as,

Q: Show that for f(n) = n³ - (ri'log, n) + 2n E O(n³ ) A give the constants: (c, n) B:arguing that your…

A: When a function is bounded above by another function with a factor asymptotically, it is referred to…

Q: Jefine f RR by f(x) = 1 if x is rational, f(x) = 0 if x is irrational. Prove that there exists a…

A:

Q: Let R+ = {x € R|x>0} be the set of positive real numbers. Let A = [a,b), a,beRb +00, be an (right…

A:

Q: 1 The domain of h(x) = - + 7 consists of all real numbers except 0 and 3, represented in interval…

A: Consider a function fx. The domain of a function fx is those values of x for which the given…

Q: a? - 1 x² + 2x+- at -5' x+2 ;x>0 Let f(x) = { Find all possible real values of a such that f(x)…

A:

Q: If h is defined by  h(x) = |X| + 2, What is the range of h if the domain of h is the interval…

A: The given function is h(x)=x+2

Q: Given a function f: R → P(Q), where R is the set of real numbers and P(Q) is the power set of…

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Q: a. Use the fact that every nonempty interval of real numbers contains both rational and irrational…

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Q: Given f(x, y) and y = y(x), we can define a composite functiong(x) = f(x,y(x)).

A: a)Given , gx=fx,yx Then, g'x=∂f∂x+∂f∂yddxyx  [∵y=yx]⇒g'x=∂f∂x+∂f∂yy'x

Q: Which of the following statements is FALSE?  A) All polynomial functions are continuous everywhere.…

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Q: Is it true or fase ? There is a number c in the interval (a, b) such that f(c) = 0.

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Q: Prove that the product xy of two real numbers x and y is nonnegative if and only if the absolute…

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Q: If b is a positive real number with b ≠1, then for all positive real numbers x and y we find

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Q: Find the number that provides a counterexample to show that the given statement is false.    For…

A: Given: For all numbers x, x&gt;1 To Find: The number that provides a counterexample to show that the…

Q: (a) Find the domain of f(x). Ⓡ O x is all real numbers except x = -4. O x is all real numbers except…

A: Given graph is  To find: a) Domain of fx.              b) f0              c) Estimate x such that…

Q: 1. ƒ : R → R, such that f(x) = 10 - 3x. 2. f: ƒ : R_ → R, such that f(x) = x - 2. Here, R denotes…

A: Given: The two-real valued functions, 1. f:ℝ→ℝ, such that f(x)=10-34x. 2. f:ℝ_→ℝ, such that…

Q: For which positive numbers a is it true that a" > 1 + x for all x?

A:

Q: (2) If h is defined by h(x) = |x| + 2, (a) What is the range of h if the domain of h is the interval…

A: Range of modules function

Q: f(x) = x+3x−2x+3x−2 and g(x) = x−1x+1x−1x+1 . The domain of f ∘∘ g is the set of all real numbers…

A: Use fog(x) =f(g(x))

Q: 1. Let x, y, and z be real numbers. Prove the following: [a] -(-x)= x [b] (-x )y = -(xy) and (- xX-…

A: Here you go. Just use a square property and answer follows.

Q: 2. Prove that a function which is analytic in the whole plane and satisfies an inequality |f(2)| <…

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Q: 1 -Find all the points where f(2)is analytic. 2+3 Let /(2) =

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Q: Let x +1 be a real number. Prove that for all n E N, x" – 1 - + x" +...+ x2 + x + 1. x – 1

A:

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.