For each relation below, determine the following.(i) Is it a function? If not, explain why not and stop. Otherwise, answer part (ii).(ii) What are its domain and image?
(a){(x, y) :x, y∈Z, y- 2x}.
(b){(x, y) :x, y∈Z, xy- 0}.
(c){(x, y) :x, y∈Z, y-x2}.
(d){(x, y) :x, y∈Z, x|y}.
(e){(x, y) :x, y∈Z, x+y= 0}.
(f){(x, y) :x, y∈R, x2+y2= 1}.
(a) {(x,y) : x, y in Z, y = 2x}
It is a function.
The domain is, Z
The image is, 2Z = { 2k : k is in Z } = set of all even integers.
(b) {(x,y) : x, y in Z, xy = 0} = {(x,y) : x, y in Z, either x = 0 or y = 0} = (X-axis) U (Y-axis)
It is not a function, since, for example, if x = 0, we can have different values of y, say, if x = 0, y can be any value in Z. So, it is a relation but not a function.
(c) {(x,y) : x, y in Z, y = x² }
It is a function.
The domain is, Z
The image is the set of all square integers, i.e. { 0, 1, 4, 9, 16, 25, 36, 49, 64, ... } = { k² : k is in Z }
(d) {(x,y) : x, y in Z, x | y }
It is not a function, since, (2,4) & (2,6) belongs to the set among others.
So, the integer has at least two images, which can't happen in a function.
It is a relation, however.
(e) {(x,y) : x, y in Z, x+y = 0, i.e. y = - x }
It is a function.
The domain is, Z
The image is Z
(f) {(x,y) : x, y in Z , x² + y² = 1 }
It is not a function, since, for, x = 0, we have, y = +1 as well as y = - 1
So (0,1) & (0,-1) both are in the set.
So, it is not a function.
For each relation below, determine the following.(i) Is it a function? If not, explain why not...
67. Explain why or why not Determine whether the following state- ments are true and give an explanation or counterexample. a. If f(x, y) x2 + y2 - 10, then Vf(x, y) = 2x + 2y b. Because the gradient gives the direction of maximum increase of a function, the gradient is always positive. c. The gradient of f(x, y, z) = 1 + xyz has four components d. If f(x, y, z) = 4, then Vf = 0
All of 10 questions, please. 1. Find and classify all the critical points of the function. f(x,y) - x2(y - 2) - y2 » 2. Evaluate the integral. 3. Determine the volume of the solid that is inside the cylinder x2 + y2- 16 below z-2x2 + 2y2 and above the xy - plane. 4. Determine the surface area of the portion of 2x + 3y + 6z - 9 that is in the 1st octant. » 5. Evaluate JSxz...
By inspecting a graph of y = sin x, determine whether the function y = sin x is increasing or decreasing on the interval The function y=sin x is on the interval By inspecting a graph of y = sin x, determine whether the graph is concave up or concave down on the interval (1,21). The graph is on the interval (1,21). Minimize f(x,y)= x2 + xy + y2 subject to y = 20 without using the method of Lagrange...
11.1) a) Verify that the function f(x,y) given below is a joint density function for r and y: ſ4.ty if 0 <r<1, 0 <y<1 f(x, y) = { 10 otherwise b) For the probability density function above, find the probability that r is greater than 1/2 and y is less than 1/3. 11.2) For the same probability density function f(x,y) as from Problem #1. Find the expected values of r and y. 11.3) a) Let R= [0,5] x [0,2]. For...
Hw2 Q1 Show that the function f(z) = z2 + z is analytic. Also find its derivative. (Hint: check CR Equations for Analyticity, and then proceed finding the derivative as shown in video 8 by any of the two rules shown in video 7] Q2 Verify that the following functions are harmonic i. u = x2 - y2 + 2x - y. ii. v=e* cos y. Q3 Verify that the given function is harmonic, and find the harmonic conjugate function...
(1 point) (a) Show that each of the vector fields F-4yi + 4x j, G-i ЗУ x2+y2 x?+yi J, and j are gradient vector fields on some domain (not necessarily the whole plane) x2+y2 by finding a potential function for each. For F, a potential function is f(x, y) - For G, a potential function is g(x, y) - For H, a potential function is h(x, y) (b) Find the line integrals of F, G, H around the curve C...
(1) For each of the following relations on R, is the relation reflexive? Is it symmetric? Is it transitive? (a)r1={(x, y)∈ R × R | xy= 0} (b) r2={(x, y)∈R×R|x2+y2= 1} (c)r3={(x, y)∈R×R||x−y|<5}
solution to 1through 10 EXERCISES domain, explain In each of the following cases, a region D is defined. Tell whether the region is ad egion is a domain it is a domain, determine whether or not it is simply connected. If it is not a doeiomain.It why not. 1. The region of definition of a magnetic field due to a steady current flowing axis li.e, the region consisting of all points (x.y,z) such that x+y0 2. The region of definition...
NO.25 in 16.7 and NO.12 in 16.9 please. For the vector fied than the vecto and outgoing arrows. Her can use the formula for F to confirm t n rigtppors that the veciors that end near P, are shorter rs that start near p, İhus the net aow is outward near Pi, so div F(P) > 0 Pi is a source. Near Pa, on the other hand, the incoming arrows are longer than the e the net flow is inward,...
9. For each of the following, provide a suitable example, or else explain why no such example exists. [2 marks each]. a) A function f : C+C that is differentiable only on the line y = x. b) A function f :C+C that is analytic only on the line y = x. c) A non-constant, bounded, analytic function f with domain A = {z | Re(z) > 0} (i.e., the right half-plane). d) A Möbius transformation mapping the real axis...