Exercise 3.39. • Given real numbers a, b, and 8, prove that the translation function f:(a,b)...
Exercise 6 requires using Exercises 4 and 5. Exercise 4. Let a be any real number. Prove that the Euclidean translation Ta given by Ta(x, y)(a, y) is a hyperbolic rigid motion. *Exercise 5. Let a be a positive real number. Prove that the transformation fa: HH given by fa(x, y) (ax, ay) is a hyperbolic rigid motion Exercise 6. Prove that given any two points P and Q in H, there exists a hyperbolic rigid motion f with f(P)...
be a coordinate function for 1.) (Coordinate functions) Let f: R a (a) (Exercise 3A) Prove that -f is a coordinate function as well. (5 (b) For which real numbers a, b e R is a f+b a coordinate function? (c) Let g:E → R be a coordinate function. Prove that there exists a line ( points) Justify your answer with a proof. (10 points) real number b E R, such that g-f+b or gf+b. (10 points)
Suppose f, g are two functions mapping positive real numbers to positive real numbers and f = O(g). Prove why each statement is true or false. (a) log2 f = O(log2 g) (b) √f = O(f) (c) fk + 100fk−1 = O(gk), for k ≥ 1
Let f: [a, b] → [a,b] be a continuous function, where a, b are real numbers with a < b. Show that f has a fixed point (i.e., there exists x e [a, b] such that f(x) = x).
Problem 3: Let f(x) be a function on the set of real numbers r > 1. Define the function g(x) for x by 1 g(s)-Σf(r/n). 1<nsr Prove that f(s) -Σμ(n)g (r/n). = 1nsz Here is the Möbius function
Suppose that the function f is defined, for all real numbers, as follows. f(x) = x-2 ifx#2 4 if x=2 Find f(-3), f(2), and f(5). s(-3) = 0 s(2) = 0 r(s) = 1 Suppose that the function g is defined, for all real numbers, as follows. if x -2 8(x)= 1-4 if x=-2 Find g(-5), g(-2), and g(4). $(-5) = 0 DO s(-2) = 1 8(4) = 1
Exercise 5. Prove that if f is a continuous and positive function on (0,1], there exists 8 >0 such that f(x) > 8 for any x € [0,1].
(Exercise 5.8 of the text) If f is a real-valued function on (a, b) and c E (a, b), we say that f is strictly increasing at c if there exists δ 〉 0 such that if c-5 〈 x 〈 c, then f(x) 〈 f(c) and if c 〈 x 〈 c+6, then f(c) 〈 f(x). We say that f is strictly increasing on (a, b) if whenever x, y E (a, b) with x 〈 y, we have...
1. Suppose that a function f, defined for all real numbers, satisfies the property that, for all x and y, f(x + y) = f(x) + f(y). (*) (a) (2 points) Name a function that satisfies property (*). Name another that doesn't. Justify your answers. (b) (3 points) Prove that any function that satisfies property (*) also satisfies f(3x) = 3f(x). (c) (5 points) Prove that any function that satisfies property (*) also satisfies f(x - y) = f(x) –...
1. Suppose that a function f, defined for all real numbers, satisfies the property that, for all x and y, f(x + y) = f(x) + f(y). (*) (a) (2 points) Name a function that satisfies property (*). Name another that doesn't. Justify your answers. (b) (3 points) Prove that any function that satisfies property (*) also satisfies f(3x) = 3f(x). (c) (5 points) Prove that any function that satisfies property (*) also satisfies f(x - y) = f(x) –...