7. Define a function G: R XRRXR as follows: G(x,y) = (2y, -x) for all (x,y) ERXR. a. Prove that G is one-to-one. [4 points) b. Prove that G is onto. [4 points)
(4) Define the function f : R -»R* by x-1/2 r> 0 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I = [0, 1] and compute the value of f du (4) Define the function f : R -»R* by x-1/2 r> 0 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I...
(4) Define the function f : R -> R* by ,--1/2 f(x) x< 0. +oo, |(a) Prove that f is measurable (with respect to the Lebesgue measurable sets). (b) Prove that f is integrable on I 0, 1and compute the value of = f du (4) Define the function f : R -> R* by ,--1/2 f(x) x
(4) Define the function f : R -> R* by .-1/2 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I [0, 1 and compute the value of f du (4) Define the function f : R -> R* by .-1/2 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I [0, 1 and...
(9) Let E R" and let A E L(R"). Define a map f : R" -> R" by f (x) A,)v. Here (is the Euclidean inner product (a) Prove that f is a C1 map and find f'(x) (b) Prove that there exist two that f U V is a bijection on R" neighborhoods of the origin in R", U and V, such (9) Let E R" and let A E L(R"). Define a map f : R" -> R"...
4. Define the function f: 0,00) +R by the formula f(x) = dt. +1 Comment: The integrand does not have a closed form anti-derivative, so do not try to answer the following questions by computing an anti-derivative. Use some properties that we learned. (a) (4 points). Prove that f(x) > 0 for all x > 0, hence f: (0,00) + (0,0). (b) (4 points). Prove that f is injective. (c) (6 points). Prove that f: (0,00) (0,00) is not surjective,...
Define the function f : Rf3 ! Rf5 by f(x)= 5x/x-3Prove: f is surjective ("onto" R\5). R {5} by 7. (15 pts) Define the function f : R\{3} f(x) = 0 Prove: f is surjective ("onto" R\{5}). I
3. (a) Let f be an infinitely differentiable function on R and define х F(x) = e-y f(y) dy. Find and prove a formula for F(n), the nth derivative of F. (b) Show that if f is a polynomial then there exists a constant C such that F(n)(x) = Cem for sufficiently large n. Find the least n for which it is true.
5. Let A = P(R). Define f : R → A by the formula f(x) = {y E RIy2 < x). (a) Find f(2). (b) Is f injective, surjective, both (bijective), or neither? Z given by f(u)n+l, ifn is even n - 3, if n is odd 6. Consider the function f : Z → Z given by f(n) = (a) Is f injective? Prove your answer. (b) Is f surjective? Prove your answer
x y be subgausure of a define a function m: XxY6 by m(x, y) : xey 1. Let (, ,y), la, y eXxY and let f... m (m(x, y)) Construct a natural bejection from F, F 2 Construct a bijection from (x nyl to the fibre Assuming 6 is finite, 1x.y = Puove 1x.ly ; Il indicates, F is Net the fire of in through (x, y)