6. Use the fact that every odd degree polynomial with real coefficients has a real root to show that the function f : R –R defined by ƒ (x) = x³ – 2.x² + x for x ER is onto. Is f one-to-one? Bonus: Let A = {a,b, c} be a set and P(A) be the power set of A. Define f : P(A) → Z by the rule f (A) = number of elements of A. Is f one-to-one? If so, prove it. Otherwise, give an counterexample.

Q: Suppose f: R → R is the squaring function, that is, f(x) = x² for every real number . Mark all of…

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Q: Q2: (a) Determine which of these functions are injective, surjective, and bijective. The function f…

A: a:Given are the following functions:i fx=2x+1ii fx=exiii fx=x3-3xTo find that the following…

Q: Viewed as a function f: R → R, a polynomial of odd degree is always surjective. True False

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Q: Let f and g be the functions defined by f : R → R+ ∪ {0} with f (x) = x2 and g : R+ ∪ {0} → R with…

A: (fog)(x) is the value of the function f(x) at g(x) (fog)(x)= f(g(x))

Q: 2. Let R be a U.F.D. and f (x), g (x) e R [x]. If f(x) g (x) is a primitive polynomial, then f (x)…

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Q: Obtain the Lagrange interpolating polynomial for a certain function f of which we know that: f (-1)…

A: Lagrange interpolating polynomial: The Lagrange interpolating polynomial is a degree polynomial that…

Q: d) Let the set A = {1, 2, 3,4} and let P(A) denote the power-set of A. Consider the following…

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Q: let f(z) be polynomial of degree at least two. show that |f(z)|>|z| for large |Z|.

A: We prove this result by contradiction.

Q: Let A = {1,2,3}, B B = = Define the functions f: AC and g: B → C by f(1) = 7, f(2)= 9, f(3) = 7,…

A: We want g o h = f.     So, we want g o h(1) = f(1) = 7            and g o h(3) = f(3) = 7   Now,…

Q: let a= {0,4,9,1.6} , b={0,3,6,12,20} and f a function. from a to b with f(x)=√x + x. A. Find f(0),…

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Q: Q1. Let f: X-Y and g: Y - Z be two functions then a. If gof: X-Z bijective function then / and g are…

A: Step 1: Step 2: Step 3: Step 4:

Q: The following questions pertain to the use of piecewise polynomials for interpolating a function f…

A: As per the question, we consider constructing a piecewise polynomial P(x) that matches a function f…

Q: 3. Let f : R→R be a function defined by f(r) = 2x + 1. Is this function (only) injective, (only)…

A: Let us consider the function f: A→B then the function is said to be  Injective or one-one :That is…

Q: Suppose that the function f(x)  is continuous everywhere. Suppose further that the points (Xj, Yj),…

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Q: We know that 1 is a zero of the function f(x) = x3 + x2 - 5x+3 the multiplicity of f at 1 is: Select…

A: We are given that 1 is a zero of f(x)=x3+x2-5x+3 ⇒(x-1) is a factor of f(x)=x3+x2-5x+3 Now, By long…

Q: Given function p :x→3- 2x. If q is another function in such that pq : x→ 1- 5x², determine the…

A: q°p= qop(x)

Q: Let f be some continuous function on an open interval I containing x0, x1 and x2, where xo <x1 <x2.…

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Q: Define f : R → R and g : R → R by the rules f(x) = 3.x – 2 for all x E R and g(x)= x² +3 for all x E…

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Q: 3. Let x0, x₁,...,xn be distinct points on the interval [a, b], where Xo <x₁ < x₂ <... < xn and let…

A:

Q: Select the one true statement. Consider a C function f: R" → R and two multi-indices a,

A: To solve this problem I will use the fact that every polynomial in any variable is smooth function.

Q: Let T„(x) be the Maclaurin polynomial of a function f(x) satisfying f(5) (x) < 1 for all x. Which of…

A:

Q: ) Prove that, if a and b are nonnegative integers such that a =4 b, then 3ª =5 3b. Using (a),…

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Q: (a) Prove that, if a and b are nonnegative integers such that a = 4 b, then 3ª =5 3b. (b) Using (a),…

A: (a) Let a ≡4 b          =&gt; 4 | (a - b)          =&gt; a - b = 4k for some integer k.…

Q: 8. Let f be an entire function such that f(z)| < A + B|z|" for all z E C, where A and B are positive…

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Q: Let P(z) = z" +am-12"-1 + ... + a1z + ao be a polynomial of degree n 2 1 with a; # 0 for at %3D…

A: We will solve Q9 as mentioned. Given Pz=zn+an-1zn-1+…+a1z+a0 be a polynomial of degree n≥1 with ai≠0…

Q: Given x = {5,10,15,20,25,30} and y function from X to Y. Let f(5) = 400, f(10) = 100, f(15) = 500, f…

A: x=5, 10, 15, 20, 25, 30  &amp;  y=100, 200, 300, 400, 500, 600. f5=400,  f10=100,  f15=500,…

Q: 2 (a) Prove that, if a and b are nonnegative integers such that a 4 b, then 3ª = 3. (b) Using (a),…

A: Given a, b are non negative integers such that a≡4b . Then to show 3a≡53b. To show the function…

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: D. Let function g: R → R be defined by g(x) = 2x + 1 5 cx in its domain. Why does your choice of c…

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Q: Show that f (a) = is Libschitz function on r+ 2

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Q: This problem is about giving an example of a function from Z to Z that is onto, but not one-to-one.…

A: