Suppose that f(i) is a periodic function of period 2a with
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- 2. Definition: If f(x) is a function, then we say that a value u is a fixed point of f(x) if and only if f (u) = u. Suppose F(x) is a given continuous function and a # 0 is a given real number. Show that u is a zero of F(x) if and only if u is a fixed point of f (x) f (x) = x + a F (x). b. Suppose F '(x) is continuous, u is a zero of F, and F'(u) ± 0. Define f (x) = x + aF(x). Prove there are values of a + 0 and ɛ > 0 so that if uo E (u – E, u + ɛ) and un+1 = f (Un) for n = Hint: Jun+1 – u| = \f (un) – f (u)]. Use the definition of f (x) and the mean а. = 0,1,2, ... then un → u as n → ∞. | value theorem.arrow_forwardQl: The Collatz conjecture function is defined for a positive integer m as follows. (COO1) g(m) = 3m+1 if m is odd = m/2 if m is even =1 if m=1 The repeated application of the Collatz conjecture function, as follows: g(n), g(g(n)), g(g(g(n))), ... e.g. If m=17, the sequence is 1. g(17) = 52 2. g(52) = 26 3. g(26) = 13 4. g(13) = 40 5. g(40) = 20 6. g(20) = 10 7. g(10) = 5 8. g(5) = 16 9. g(16) = 8 10. g(8) = 4 11. g(4) = 2 12. g(2) = 1 Thus if m=17, apply the function 12 times in order to reach m=1. Use Recursive Function.arrow_forwardFor all a and b in the domain of a function f, the function is injective iff f(a) #f(b) a=Db f(a)=f(b) a=b f(a)=f(b) atb o fly)=x iff f(x)=y A Moving to another question will save this response. 5arrow_forward
- Define functions f, g, h, all of which have R as their domain and R as their target. f(x) = 3x + 1 g(x) = x² h(x) = 2x What is (g o h)(5)? What is (f o go h)(1)? What is (f of ¹)(15)? 22arrow_forwardSuppose that the equation ax b .mod n/ is solvable (that is, d j b, whered D gcd.a; n/) and that x0 is any solution to this equation. Then, this equation has exactly d distinct solutions, modulo n, given by xi D x0 C i.n=d / fori D 0; 1; : : : ; d 1arrow_forwardQ. Let A = {a, b, c, d, e} and B = {1, 2, 3, 4, 5, 6, 7, 8}. How many functions f : A → B(a) ... are injective?(b) ... are not injective?(c) ... are such that f(a) = f(b) = f(c)?(d) ... are such that exactly three elements of A have 8 as an image?(e) ... are surjective?arrow_forward
- A quadratic function is a function of the form f(x) = ax² + bx+c, where a, b, c ER are fixed constants. Define a function quad_coefficients which accepts a quadratic function fas input, and returns a tuple containing the value a at index 0, b at index 1, and c at index 2.arrow_forward6 For the following pairs of functions, first decide whether f(n) dominates g(n), or g(n) dominates f(n); then decide whether f = 0(g), or f = N(g), or f = 0(g), and briefly explains. i. f(n) = n²,g(n) = 1000n + 30 ii. f(n) = /n, g(n) = n} iii. f(n) = 10log2 n, g(n) = log10(n³) iv. f(n) = n100, g(n) = 1.2".arrow_forwardLet A = {1, 2, 3, 4} and B = {a, b, c}. Give an example of a function f: A -> B that is neither injective nor surjective.arrow_forward
- A function is said to be if and only if f(a) = f(b) implies that a = b for all a %3D and b in the domain of f. O a. No answer is correct O b. One to One Oc. One to many O d. Many to onearrow_forward3. (a) Consider the following algorithm. Input: Integers n and a such that n 20 and a > 1. (1) If 0arrow_forwardA service station has both self-service and full-service islands. On each island, there is a single regular unleaded pump with two hoses. Let X denote the number of hoses being used on the self-service island at a particular time, and let Y denote the number of hoses on the full-service island in use at that time. The joint pmf of X and Y appears in the accompanying tabulation. p(x, y) 0 1 2 0 0.10 0.05 0.01 x 1 0.06 0.20 0.07 2 0.06 0.14 0.31 (a) What is P(X = 1 and Y = 1)? P(X = 1 and Y = 1) = | (b) Compute P(X ≤ 1 and Y < 1). P(X ≤1 and Y < 1) = (c) Give a word description of the event { X0 and Y + 0 }. One hose is in use on one island. One hose is in use on both islands. At least one hose is in use at both islands. At most one hose is in use at both islands. Compute the probability of this event. P(X +0 and Y 0) = [ (d) Compute the marginal pmf of X. x 0 Px(x) Compute the marginal pmf of Y. y Py(y) 0 Using Px(x), what is P(X ≤ 1)? P(X ≤ 1) = = 1 2 1 2arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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