. 4 points) Let f(x) = Anx" + an-12-1 +ajx + do with ao, a1,..., an...
[4 points) Let f(x) = anx" + an-12-1...+91x + ao with ao, a1,..., an being real numbers. Prove that f(x) E O(x") by finding a pair of witnesses C and k such that f(x) < C|x"| whenever >k.
I need help with my discrete math question. thanks in advance Let f(x) = 0 + 0,-1-1...+ar+ao with 00, 01,..., an being real numbers. Prove that f(0) E O(") by finding a pair of witnesses C and k such that f(x) < Cx" whenever I k.
1. Let f(3) = 20.3" +ajx"-1 + ... + an, where ao, ar- an are integers. Show that if d consecutive values of f (i.e., values for consecutive integers) are all divisible by the integer d, then d f (x) for all integers r.
(1) Let a (.. ,a-2, a-1,ao, a1, a2,...) be a sequence of real numbers so that f(n) an. (We may equivalently write a = (abez) Consider the homogeneous linear recurrence p(A)/(n) = (A2-A-1)/(n) = 0. (a) Show ak-2-ak-ak-1 for all k z. (b) When we let ao 0 and a 1 we arrive at our usual Fibonacci numbers, f However, given the result from (a) we many consider f-k where k0. Using the Principle of Strong Mathematical Induction slow j-,-(-1...
1.(1) Let A={f(x): f(x)-axx? +ajx + ap} where a, eR (i=1,2,3). Define f+g by (f+g)(x)=(a+b)x² + (a1 +b ) x + (ao+b) also define (rf)(x)=(ra) x? +(ra)x+rao Show that A is vector space.
Problem (5), 10 points Let a0:a1, a2, be a sequence of positive integers for which ao-1, and a2n2an an+ for n 2 0. Prove that an and an+l are relatively prime for every non-negative integer n. 2n+an for n >0 Problem (5), 10 points Let a0:a1, a2, be a sequence of positive integers for which ao-1, and a2n2an an+ for n 2 0. Prove that an and an+l are relatively prime for every non-negative integer n. 2n+an for n >0
1. (15 points) Let F be a field and let f(x) = ama + an–1.29-1 +...+azur? + a2x2 +212 + a0 € F[x]. Show that I 1 is a factor of f(x) if and only if ao + a1 + a2 + 03 +... + n = 0
(-/4 Points] DETAILS GHCOLALG12 2.1.058. Let the function be defined by y = f(x), where x and f(x) are real numbers. Find (2), R-5), k), and f(2-1). f(x) = 5x + 7 R(2) f(-5) = f(K) 10 (k2 - 1) =
(c) Let f : IR2 -R2 be given by f(x,)= (a1)2-y1, (-12) Let S, S' be the subsets of R2 as indicated in the picture below. Prove that f maps S onto S' (0,1) v-axis V=1 (2,1) (1,1) y =(x-1)2 у-ахis u 1 v=u-1 u-axis (1,0) (0,0) х-аxis (1,0) (c) Let f : IR2 -R2 be given by f(x,)= (a1)2-y1, (-12) Let S, S' be the subsets of R2 as indicated in the picture below. Prove that f maps S...
Problem 3 (7 points) Define three functions A,A,As as follows: βί(x) = 0 whenever x < 0 and A(x)-1 whenever x > 0, Moreover we let A (0)-0, β2(0)-1 and A3(0) . Let f be a bounded function on [-1, 1] (a) Prove that f ER(B1) if and only if lim0+ f(x)-f(0). In this case prove that 1 『(0) elf, -1 (b) State and prove a similar result for A. (c) Prove that f ER(B3) if and only if f...