4. Prove that the function that sends an ordered pair (p,q) of elements in P(R) to* p(x) q(x)e da defines an inner product on P(R).0

Please refer to the explanation.Explanation:Step 1:Step 2:Step 3:Step 4:Step 5:

Answered: 4. Prove that the function that sends…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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f: X→ Y and g: Y→V be bijective functions. Show the following: ) gof: X → V is injective. ) gof: X→ V is surjective. ) (gof)-¹ = f¹og-¹ v injective if x1 #xn → v(x1) = v(xn)
6. Give an example of a polynomial function that has each of the following end behaviours: a) As x→- ∞, y→-0 and as x→ 0, y→*. b) As x ± 0,y→ 0. c) As x+∞, y→ -. d) As x -∞, y→ 0 and as x→0, y→ -∞,
1. Determine whether the following polynomials span P. =1+x, p, = 1 – x, P2 P3D1+x+x², P, = 2-x² 3D2 3:28 PM 豆 IA 2/23/2022 e Type here to search WQHD | 165HZ NOG ZEPHYRUS 100% DC-P3 PANTONE Validated F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 ESC E3 O/京 DELETE AURA 23 24 4 @ % & 7 { } TAB Q W E T Y P. [ 1O 15 林

Ans. no. 4 only.
Which of the following function f: R X R → R is not onto? 1 2x - 3 2 3x3 - x 3 x2-2x 4 f(x) = x2- 5x + 5
Let V be the set of functions f: R → R. For any two functions f, g in V, define the sum f + g to be the function given by (f + g)(x) = f(x) + g(x) for all real numbers. For any real number c and any function f in V, define scalar multiplication cf by (cf)(x) = cf(x) for all real numbers . Answer the following questions as partial verification that V is a vector space. (Addition is commutative:) Let f and g be any vectors in V. Then f(x) + g(x). since adding the real numbers f(x) and g(x) is a commutative operation. (A zero vector exists:) The zero vector in V is the function f given by f(x) = for all. (Additive inverses exist:) The additive inverse of the function f in V is a function g that satisfies f(x) + g(x) = 0 for all real numbers . The additive inverse of f is the function g() for all . me for all real numbers a (Scalar multiplication distributes over vector addition:) If c is any real number and fand g are two vectors in V, then c(f+g)(x) = c(f(x) + g(x)) =
13. (9 points) Let D be the set of finite subsets of positive integers. Let S be the set of all positive integers greate than or equal to 2. Define a function T:S → D as follows: For each integer n ≥ 2, T(n) = the set of all even factors of n. a) Find T(10). b) Find T(17) c) Find T(m), where m is any odd positive integer.
Suppose f: R → R is defined by the property that f(x) = x + x² + x³ for every real number x, and g: R → R has the property that (gof)(x) = = x for every real number a. Then g" (0) = 1/2 1 ○ 1/6

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4. Prove that the function that sends an ordered pair (p,q) of elements in P(R) to * p(x) q(x)e da defines an inner product on P(R). 0