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6041/6431 Spring 08 Quiz 2 Wednesday, April 16, 730 930 PM SOLUTIONS Name Recitation Instructor TA Question Part If A = (1,2),(4,3) and f(x) = 2x3 4x 5, find f(A) Welcome to Sarthaks eConnect A unique platform where students can interact with teachers/experts/students to get solutions to their queries Chapter 2 (maths 3) 1 CHAPTER 2 FOURIER SERIESPERIODIC FUNCTIONS A function f (x ) is said to have a period T if for all x, f ( x T ) = f ( x) , where T is apositive constant

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If f(x)=4^x/4^x 2 then f(1/50) f(2/50)-P X > 1/2 = 1 −P X ≤ 1/2 = 1−FX (1/2) = 1 −3/4 = 1/4 (2) (b) This is a little trickier than it should be Being careful, we can write P −1/2 ≤ X < 3/4 = P −1/2 < X ≤ 3/4P X = −1/2 −P X = 3/4 (3) Since the CDF of X is a continuous function, the probability that X takes on any specific value is zero This implies0 1, C x y2 y x f x y a) Find the value of C that would make f x, a valid probability density function y b) Find the marginal probability density function of X, f X (x) c) Find the marginal probability density function of Y, f Y (y) d) Find P (X > 2 Y) e) Find P (X Y < 1) f) Are X and Y independent?

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 Example 10 Show that the function f N N, given by f (1) = f (2) = 1 and f (x) = x 1, for every x > 2, is onto but not oneone Here, f(x) = 1 for =1 1 for =2 1 for >2 Here, f (1) = 1 f (2) = 1 Check onto f N N f(x) = 1 for =1 1 for =2 1 for >2 Let f(x) = y , such that y N Here, y is a natural number & for every y, there is a value of xLet f N → N be a function defined as f (x) = 4 x 2 1 2 x 1 5 Show that f is not invertible, but f N → S invertible (whereS is range of f) Find the inverses f and hence find f − 1 ( 3 1 ) and f − 1 ( 8 7 )Then xn → 0 as n → ∞, while if x = 1, then xn → 1 as n → ∞ So fn → f pointwise where f(x) = {0 if 0 ≤ x < 1, 1 if x = 1 Although each fn is continuous on 0,1, their pointwise limit f is not (it is discontinuous at 1) Thus, pointwise convergence does not, in general, preserve continuity Example 54 Define fn 0,1 → R

Math 115 HW #5 Solutions From §129 4 Find the power series representation for the function f(x) = 3 1−x4 and determine the interval of convergenceYour input find all numbers $$$ c $$$ (with steps shown) to satisfy the conclusions of the Mean Value Theorem for the function $$$ f=x^{3} 2 x $$$ on the interval $$$ \left10, 10\right $$$ The Mean Value Theorem states that for a continuous and differentiable function $$$ f(x) $$$ on the interval $$$ a,b $$$ there exists such number $$$ c $$$ from that interval, that $$$ fFree functions calculator explore function domain, range, intercepts, extreme points and asymptotes stepbystep

Math 113 HW #9 Solutions 1 Exercise 4150 Find the absolute maximum and absolute minimum values of f(x) = x3 −6x2 9x2 on the interval −1,4 Answer First, we find the critical points of f Add 1 to both sides y 1 = 4x Divide both sides by 4 y 1 4 = x Where ever there is a x write y and wherever there is a y write x y 1 4 = x is changed to x 1 4 = y = f −1(x) f −1(x) = x 1 4 Answer link Ex 51, 34 Find all the points of discontinuity of f defined by 𝑓(𝑥)= 𝑥 – 𝑥1Given 𝑓(𝑥)= 𝑥 – 𝑥1 Here, we have 2 critical points x = 0 and x 1 = 0 ie x = 0, and x = −1 So, our intervals will be When 𝒙≤−𝟏 When −𝟏

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Find f (–1)" (pronounced as "fofx equals 2x plus three;If f (x) = 4 x 2 4 x , f (x) f (1 − x) = a and f (9 7 1 ) f (9 7 2 ) ⋯ f (9 7 9 6 ) = b then order pair (a, b) is This question has multiple correct options A2 If a < b, then F(a) ≤ F(b) for any real numbers a and b 163

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The function f is defined by f (x) = x4 −4x2 x 1 for −5 ≤ x ≤ 5 What is the interval in which the minimum of value of f Purely a graphical approximation;F(x) = x^2 4x for x less than or equal to 2, find f^1(x) y^2 4y = x, y less than or equal to 2 y^2 4y )x) = 0 y = 4 plus minus squareroot 4(1) (x)/2 = 2 plus minus squareroot 4 x, y less than or equal to 2 f^1 (x) = 2 squareroot 4 xExtended Keyboard Examples Upload Random Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals For math, science, nutrition, history, geography, engineering, mathematics, linguistics, sports, finance, music

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F(x) = (c2x c x 1 cx x x>1 Solution The partial functions of f(x) are continuous for x < 1 and x>1 becauseX!0 2 2 p 4 x x2 Solution lim x!0 2 p 4 x2 x 2 = lim x!0 2 p 4 2x x 2 p 4 x2 2 p 4 x2 = lim x!0 4 (4 x2) x2(2 p 4 x2) = lim x!0 1 2 p 4 x2 = 1 4 3 For which value of the constant cis the function f(x) continuous on (1 ;1)?Solve your math problems using our free math solver with stepbystep solutions Our math solver supports basic math, prealgebra, algebra, trigonometry, calculus and more

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The statement "if f(x) = 1" tells me you are defining the function f to have the property that f(x) = 1 regardless of the value of x This is a particularly uninteresting function It is like a broken bathroom scale that always gives the same weight regardless of who stands on it Now look at f(f(x)) = 2x 4 What I see is f(f(x)) = 2x 4Sal finds the inverses of f(x)=x4 and g(x)=2x1 Sal finds the inverses of f(x)=x4 and g(x)=2x1 opposite we solve for X in terms of Y so let's subtract 4 from both sides you get Y minus 4 is equal to negative x and then to solve for x we could multiply both sides of this equation times negative 1 and so you get negative y plus 4 is9Using the lefthand Riemann sum with n= 4, approximate Z 9 1 1 x dx Answer Z 9 1 1 x dxˇ2 1 1 1 3 1 5 1 7 = 352 105 10Suppose that f(2) = 4, and that the table below gives values of f0for xin the interval 0;12

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For any set X, the identity function id X on X is surjective;Fexp( 2 1 2 t)(x) = exp(1 2 x2);x2R We will discuss this example in more detail later in this chapter We will also show that we can reinterpret De nition 1 to obtain the Fourier transform of any complex valued f 2L2(R), and that the Fourier transform is unitary on this space Theorem 3 If f;g2L2(R) then Ff;Fg 2L2(R) and Z 1 1 f(t)g(t2 (a) Define uniform continuity on R for a function f R → R (b) Suppose that f,g R → R are uniformly continuous on R (i) Prove that f g is uniformly continuous on R (ii) Give an example to show that fg need not be uniformly continuous on R Solution • (a) A function f R → R is uniformly continuous if for every ϵ > 0 there exists δ > 0 such that f(x)−f(y) < ϵ for all x

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 Relations and Functions Class 12 MCQs Questions with Answers Question 1 Question 2 Question 3 If F R → R such that f (x) = 5x 4 then which of the following is equal to f 1 (x) Question 4 Question 5 Question 6 Question 7For example, if the red and green dice show the numbers 6 and 4, then X = 6 and Y = 1 Write down a table showing the joint probability mass function for X and Y, find the marginal distribution for Y, and compute E(Y) Here is a table showing the joint probability mass function, with theFree math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with stepbystep explanations, just like a math tutor

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X π 6= x,x ∈ R, and if y > 1 then there is no x ∈ R such that y = f(x) c) The function f is injective because f(x) < f(y) if x < y,x,y ∈ R, but not surjective as a map from R to R, because there exists no x ∈ R such that f(x) = −1 4 SOLUTION SET FOR THE HOMEWORK PROBLEMSInverse\y=\frac{x}{x^26x8} inverse\f(x)=\sqrt{x3} inverse\f(x)=\cos(2x5) inverse\f(x)=\sin(3x) functioninversecalculator en Related Symbolab blog posts Functions A function basically relates an input to an output, there's an input, aFind fofnegativeone") In either notation, you do exactly the same thing you plug –1 in for x, multiply by the 2, and then add in the 3, simplifying to get a final value of 1

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F(x) = 2x^2–5x4 2(f(x) = 2(2x^2–5x4 )= 4x^2–10x8 f(2x) = 2(2x)^2–5(2x)4 = 8x^2–10x4 4x^2–10x8 = 8x^2–10x4 8x^2–10x4 4x^210x8=0 4x^2–4 = 0The function f Z → {0, 1} defined by f(n) = n mod 2 (that is, even integers are mapped to 0 and odd integers to 1) is surjective;Given f (x) = 3x 2 – x 4, find the simplified form of the following expression, and evaluate at h = 0 This isn't really a functionsoperations question, but something like this often arises in the functionsoperations context

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Now f(x,y) = f(0, w−2 3) = 3(w−2 3)2 = (w −2)2 = w Therefore f is onto R (g) f Z → Z ×Z by f(x) = (x4,x−1) • ONETOONE Let a,b ∈ Z Then f(a) = f(b) ⇒ (a4,a−1) = (b4,b−1) ⇒ a4 = b4 and a−1 = b−1 ⇒ a = b and a = b ⇒ a = b Therefore f is onetoone • ONTO COUNTEREXAMPLE There is no way to get to If f(x) = 4x3/6x4, x ≠ 2/3, then show that fof(x) = x for all x ≠ 2/3 What is the inverse of f?The function FX(x) is also called the distributionfunction of X 162 Properties of a CumulativeDistribution Function The valuesFX(X)of the distributionfunction of a discrete random variable X satisfythe conditions 1 F(∞)= 0 and F(∞)=1;

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But let's use "f" We say "f of x equals x squared" what goes into the function is put inside parentheses after the name of the function So f(x) shows us the function is called "f", and "x" goes in And we usually see what a function does with the input f(x) = x 2 shows us that function "f" takes "x" and squares itPrecalculus questions and answers;Chapter 4 Taylor Series 17 same derivative at that point a and also the same second derivative there We do both at once and define the second degree Taylor Polynomial for f (x) near the point x = a f (x) ≈ P 2(x) = f (a) f (a)(x −a) f (a) 2 (x −a)2 Check that P 2(x) has the same first and second derivative that f (x) does at the point x = a 43 Higher Order Taylor Polynomials

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To find the x x coordinate of the vertex, set the inside of the absolute value x − 4 x 4 equal to 0 0 In this case, x − 4 = 0 x 4 = 0 Add 4 4 to both sides of the equation Replace the variable x x with 4 4 in the expression Simplify ( 4) − 4 ( 4) 4Conversely, if the composition of two functions is bijective, it only follows that f is injective and g is surjective Cardinality If X and Y are finite sets, then there exists a bijection between the two sets X and Y if and only if X and Y have the same number of elements Indeed, in axiomatic set theory, this is taken as the definition of "same number of elements" (equinumerosity), andAsked in Mathematics by Afreen ( 307k points)

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 f^(1)(x) = sqrt(x4) We have f(x) = x^24 And we seek the inverse function, f^(1)(x) My preferred approach is to put y = x^24 A And rearrange to form an explicit relationship x=f(y), and this function is the inverse, f^(1)(x) So, from A we have x^2 = y4 x = sqrt(y4) Thus f^(1)(x) = sqrt(x4) Note that by the formal definition f^(1)(x) is not aMinimum f = 463, nearly This is improved to 8sd, \displaystyle {} , using an iterative numerical methodSolve for y when x = –1" Now you say "f (x) = 2x 3;

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If not, find Cov (X, Y) 8We are told that mathf(x^2 1) = x^4 5x^2 3/math Using the substitution mathu = x^2 1/math, we thus have mathf(u) = x^4 (2 3)x^2 (1 3 1 Students can solve NCERT Class 10 Maths Polynomials MCQs with Answers to know their preparation level 1 If one zero of the quadratic polynomial x² 3x k is 2, then the value of k is 2 Given that two of the zeroes of the cubic polynomial ax 3

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The function f R → R defined by f(x) = 2x 1 is surjective (and even bijective), because for every real number y, we have an x such that f(x) = y such an appropriate x is (y − 1)/2You used to say "y = 2x 3;If X takes on only a finite number of values x 1, x 2, , x n, then the distribution function is given by (5) EXAMPLE 23 (a) Find the distribution function for the random variable X of Example 22 (b) Obtain its graph (a) The distribution function is F(x) d 0 ` x 0 1 4 0 x 1 3 4 1 x 2 1 2 x ` F(x) e 0 ` x x 1 f(x 1) x 1 x x 2 f(x 1) f(x

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