Which of the following families of real functions are algebras of functions? (For each, either show it is an algebra of functions, or else identify a necessary property that is not satisfied.) i. Polynomials of degree at most 2. (That is, polynomials of the form ax2 +bx+c, where any or all of a, b, c may be 0.) ii. Functions f such that f(x) ≤ 6. iii. Functions f that are everywhere defined, and that are continuous at the point c = 2.

Which of the following families of real functions are algebras of functions? (For each, either show it is an algebra of functions, or else identify a necessary property that is not satisfied.) i. Polynomials of degree at most 2. (That is, polynomials of the form ax2 +bx+c, where any or all of a, b, c may be 0.) ii. Functions f such that f(x) ≤ 6. iii. Functions f that are everywhere defined, and that are continuous at the point c = 2.

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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D. Is the following familiy of real functions an algebras of functions? (either show it is an algebra of functions, or else identify a necessary property that is not satisfied.) - Functions f such that f(x) ≤ 6.
Please answer with explation
§ : Falg Falg is a polynomial Show that if function that is injective, then is surjective.
h4. Which of the following families of real functions are algebras of functions? (For each, either show it is an algebra of functions, or else identify a necessary property that is not satisfied.) Polynomials of degree at most 2. (That is, polynomials of the form ax 2 + bx + c , where any or all of a , b , c may be 0.)i. Functions f such that f ( x ) ≤ 6.
What 3 functions classify as polynomial functions?
Which of the following families of real functions are algebras of functions? (For each, either show it is an algebra of functions, or else identify a necessary property that is not satisfied.)i. Polynomials of degree at most 2. (That is, polynomials of the form ax2 +bx+c, where any or all of a, b, c may be 0.)ii. Functions f such that f(x) ≤ 6.iii. Functions f that are everywhere defined, and that are continuous at the point c = 2.
can you please do d ,please provide explanations
2. In each item below, you are given sets A and B, and a formula for a (non?)function f: A → B. In each case, determine whether or not the formula gives a well-defined function. Fully justify your answers. (a) A = Q = B, f(r) = a root in B of the polynomial X² - r (b) A = C = B, f(2)= a root in B of the polynomial X² – z (c) A = P(R) - {0}¹, B = R₁ ƒ(E) = the smallest element of E (d) A = P(R), B = R, ƒ(E) = the smallest element of E if E admits a smallest ele- ment, and f(E)= 0 otherwise A function f : A → B is injective (or one-to-one) if it satisfies the following condition: for all a₁, a2 € A, if f(a₁) = f(a₂), then a₁ = a₂; f is surjective (or onto) if f(A) = B, or, more explicitly, if for each b E B there exists a € A such that f(a) = b; f is bijective if f is both injective and surjective. Here are some examples. • The squaring function f : R → R given by f(x) = x² is not injective because f(-1) = 1 = f(1); f also fails to be surjective because -1 is not in the range of f. The…