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This is the Solution of Question From RD SHARMA book of CLASS 11 CHAPTER LIMITS AND DERIVATIVES This Question is also available in R S AGGARWAL book of CLASS As an example, a classic result of Ritt shows that permutable polynomials are, up to a linear homeomorphism, either both powers of x, both iterates of the same polynomial, or both Chebychev polynomials We say f and g commute (with respect to composition) The property is called "commutativity"Learn how to solve f(g(x)) by replacing the x found in the outside function f(x) by g(x)

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F x ax g x x 2-4x 9- Davneet Singh is a graduate from Indian Institute of Technology, Kanpur He has been teaching from the past 10 years He provides courses for Maths and Science at Teachoo42 JABeachy 3 b−1x −ab−2 bxa x2 1 x2 ab−1x −ab−1x 1 −ab−1x −a2b−2 1 a2b−2 Thus we get x21 = (b−1x−ab−2)(bxa)(1a2b−2)Multiplying both sides of the equation by b2 and rearranging the terms gives us (abx)(a−bx)b2(x21) = a2b2 In Z3, we must have b2 = 1, since b6= 0, and then for all possible values of awe have a2b2 6= 0, so ( a2b2)−1 exists in Z

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 43 /5 heart 50 ardni313 A function f (x) and g (x) then (f g) (x) = x² x 6 Further explanation Like the number operations we do in real numbers, operations such as addition, installation, division or multiplication can also be done on two functions Suppose a function f (x) and g (xMathf(x)=x/math Function is giving the absolute value of mathx/math whether mathx/math is positive or negative See the y axis of graph which is mathf(x)/math against mathx/math, as x axis It shows y axis values or mathf(xThere exists x 2A such that (g f)(x) = g(f(x)) = z Therefore if we let y = f(x) 2B, then g(y) = z Thus g is surjective Problem 338 In each part of the exercise, give examples of sets A;B;C and functions f A !B and g B !C satisfying the indicated properties (a) g is not injective but g f is injective

The Derivative tells us the slope of a function at any point There are rules we can follow to find many derivatives For example The slope of a constant value (like 3) is always 0;Degf(x)g(x) = deg f(x) deg g(x) In particular, this applies if Ris an integral domain Proof Write f(x) = a 0 a nxn and g(x) = b 0 b mxm, with a n;b m6= 0 Then f(x)g(x) = a 0b 0 (a 0b 1 a 1b 0)x a nb mxnm Thus the largest power of xthat can occur is xnm, so degf(x)g(x) deg f(x)deg g(x) If Rhas no zero divisors, then 12 Find the value of f(n) for n integer 13 Assume f(x) = g(x)h(x), where g(x) and h(x) have integral coe cients and degree less than 105 Look at the product of the roots of g(x) 14 Sophie Germain's Identity 15 We have that a;b;c;dare distinct roots of P(x) 5 16 One way to solve this problem is by letting A n 1 = 1x xn 1 and doing

Suppose you are given the two functions f (x) = 2x 3 and g(x) = –x 2 5Composition means that you can plug g(x) into f (x)This is written as "(f o g)(x)", which is pronounced as "fcomposeg of x"And "( f o g)(x)" means "f (g(x))"That is, you plug something in for x, then you plug that value into g, simplify, and then plug the result into fN=0 2 n 1 (2 n 1)2 1 = 1 The original Mun tz theorem (1914) gave a criterion for density in C(0;1) with respect to the supnorm In that case, we do need to include the constant functions, since all positive powers x vanish at x= 0 If ( n) 1 n=1 is a sequence of distinct positive real numbers with inf n2N n>0, then the linear span of Two functions f and g are said to be equal if f 1 the domain of f = the domain of g 2 the codomain of f = the codomain of g 3 f (x) = g (x) for all x Question 5 A function f (x) is said to be an odd function if (a) f (x) = f (x) (b) f (x) = f

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G(x) → "g of x" → "operation(s), g, applied to a given quantity or value , x" "g(x) = x1" tells you to simply subtract 1 from whatever xvalue is given g(5)= 5–1=4 g(6)= 6–1=5 g(9)= 9–1=8 g(x)=x1 f(x) → "f of x" → "operations, f, applied to a gIf F(x)=x has no real solution then also F(F(x)=x has no real solution 1 We will show that g(x) is differentiable and the derivative is g ′ (x) = f(x b) − f(x a) Our strategy is going to show that the following limit exists which will give us the derivative we wanted lim h → 0g(x h) − g(x) h = lim h → 0∫baf(x t h) − ∫baf(x t) h Using u = t h substitution, (simple stuff)

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Basic properties In the previous Lecture 17 we introduced Fourier transform and Inverse Fourier transform \begin{alignMore than just an online integral solver WolframAlpha is a great tool for calculating antiderivatives and definite integrals, double and triple integrals, and improper integrals It also shows plots, alternate forms and other relevant information to enhanceComposite functions and Evaluating functions f(x), g(x), fog(x), gof(x) Calculator 1 f(x)=2x1, g(x)=x5, Find fog(x) 2 fog(x)=(x2)/(3x), f(x)=x2, Find gof(x

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(a) For any constant k and any number c, lim x→c k = k (b) For any number c, lim x→c x = c THEOREM 1 Let f D → R and let c be an accumulation point of D Then lim x→c f(x)=L if and only if for every sequence {sn} in D such that sn → c, sn 6=c for all n, f(sn) → L Proof Suppose that lim x→c f(x)=LLet {sn} be a sequence in D which converges toc, sn 6=c for all nLet >0And so on Here are useful rules to help you work out the derivatives of many functions (with examples below)Note the little mark ' means derivative of, andFa ;x gbecause the rst position at which the string of symbols P (a ) and P (x ) di er is the position number 3 The sub expression starting from position 3 is a and x respectively Substitution and Uni cation Example Example (Disagreement Set) Find the Disagreement Set for

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Tann axdx= tann1 ax a(1 n) 2F 1 n 1 2;1;In other words, we are looking for a function g with the property that x = g (f (x)) for x in a certain domain If such a function g exists, it is called the inverse function of fFrom F to F if and only if f(x) = g(x) in Fx 44 211,15,17,27,29 suggested problems 45 Irreducibility in Qx * Thm 421 Rational Root Test Let f(x) = a nxn a n 1xn 1 a 0 be a polynomial with integer coe cient if r 6= 0 and the rational number r/s (in lowest terms) is a root of f(x) then rja 0 and sja n Ex 1 in textbook

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Tan2 ax (90) Z tan3 axdx= 1 a lncosax 1 2a sec2 ax (91) Z secxdx= lnjsecx tanxj= 2tanh 1 tan x 2 (92) Z sec2 axdx= 1 a tanax (93) Z sec3 xdx= 1 2 secxtanx 1 2 lnjsecx tanxj (94) Z secxtanxdx= secx (95) Z sec 2xtanxdx= 1 2 sec x (96) Z secn xtanxdx= 1 n secn x;n6= 0 (97) Z cscxdx= ln tan xThe slope of a line like 2x is 2, or 3x is 3 etc;This preview shows page 29 32 out of 451 pages Let us now consider two absolutely convergent Fourier series f (x) = a n e n (x), g (x) = b n e n (x) and calculate their scalar product The multiplication theorem for absolutely convergent series shows that f (x) g (x) = a p b q e p (x) e q (x) = a p b q e p − q (x) on using the relations e p (x) e q (x) = e p q (x), e n (x) = e − n (x

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N= an where nis a positive integer (this follows from rules 6 and 8) 10lim x!a n p x= n p a, where n is a positive integer and a>0 if n is even (proof needs a little extra work and the binomial theorem) 11lim x!a n p f(x) = n p lim x!af(x) assuming that the lim x!af(x) >0 if nis even (We will look at this in more detail when we get toLet f(x) be a continuous function in an interval a,b, andlet N be any number between f(a) andf(b) Thenthere is a number c in a,b such thatf(c) = N 3A preliminary result about the definite integral Theorem Let f(x) be a continuous function on the interval a,b Then there exists a c in a,b forwhich f(c) (b a) = ∫ b a f(x)dxInfinity As a Limit The most likely source for the question whether 1/0 = ∞ is a realization that dividing 1 by ever smaller ( real) numbers produces numbers arbitrary large In this context, ∞ is understood as a very big, in fact, even bigger than any other, number In a sense, this is a good idea that may be worked out rigorously

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 The simplest polynomial that can be written is h ( x) = A x ( x 1) Then, use the fact that h ( 1) = f ( 1) − 1 = 3 − 1 = 2 to determine A You can go back to f trivially then Once you have chosen a particular choice for f, finding g, as explained in the above posts, is fairly trivial (just substitute x → x − 1, expand and simplify)Problem Set 5 Solutions Sam Elder Problem 1 (3111) Let fbe a polynomial of degree n, say f(x) = P n k=0 c kx k, such that the rst and last coe cients c 0 and c n have opposite signs Prove that f(x) = 0 for at least one positive xGet stepbystep solutions from expert tutors as fast as 1530 minutes Your first 5 questions are on us!

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Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals For math, science, nutrition, historyP) if and only if f(x) = a for some x 2Z p if and only if x2 = a or x 2 a = 0 Thus, the polynomial x a has a root By the Factor theorem thats if and only if x2 a is reducible x2 a is reducible if and only if, by the factor theorem and degree formula, it has exactlyThis is key, because different transformations of a single f(x) = 0 to x = g(x) can result in a sequence of x n that diverges, converges to the root slowly, or converges to the root quickly One good way to measure the speed of convergence is to use the ratio of the errors between

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To find the answers, all I have to do is apply the operations (plus, minus, times, and divide) that they tell me to, in the order that they tell me to ( f g ) ( x) = f ( x) g ( x) = 3 x 2 4 – 5 x = 3 x 2 4 – 5 x = 3 x – 5 x 2 4 = –2 x 6 ( f – g ) ( x) = f ( x) – g ( x)Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with stepbystep explanations, just like a math tutorEvaluate f (g(x)) f ( g ( x)) by substituting in the value of g g into f f f (x3) = 2(x3)−1 f ( x 3) = 2 ( x 3) 1 Simplify each term Tap for more steps Apply the distributive property f ( x 3) = 2 x 2 ⋅ 3 − 1 f ( x 3) = 2 x 2 ⋅ 3 1 Multiply 2 2 by 3 3

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Solve your math problems using our free math solver with stepbystep solutions Our math solver supports basic math, prealgebra, algebra, trigonometry, calculus and moreIn mathematics, function composition is an operation that takes two functions f and g and produces a function h such that h(x) = g(f(x))In this operation, the function g is applied to the result of applying the function f to xThat is, the functions f X → Y and g Y → Z are composed to yield a function that maps x in X to g(f(x)) in Z Intuitively, if z is a function of y, and y is aRestriction of a convex function to a line f Rn → R is convex if and only if the function g R → R, g(t) = f(xtv), domg = {t xtv ∈ domf} is convex (in t) for any x ∈ domf, v ∈ Rn can check convexity of f by checking convexity of functions of one variable

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Assume that all the coefficients of f(x) = a n n−1 n x a n−1 ··0 are reduced mod pand also that a n ≡0 mod p By dividing out by a n, can assume that f(x) is monic (ie, highest coefficient is 1) We can also assume degree n of f is less than p If not, can divide f by x p − x to get f(x) = g(x)(x p − x) r(x That is not true in general or do you mean in some limit?F (g (2)), g (x)=2x1, f (x)=x^2 \square!

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N, then F(α)=Fα, the set of polynomials in αwith coefficients in F In fact, Fα isthesetF n−1αofallpolynomialsof degree at most n− 1withcoefficientsinF, and 1,α,,αn−1 form a basis for the vector space Fα over the field F Consequently, F(α)F=n Proof Let f(X) be any nonzero polynomial overF of degree n− 1 orIntuitively, a function is a process that associates each element of a set X, to a single element of a set Y Formally, a function f from a set X to a set Y is defined by a set G of ordered pairs (x, y) with x ∈ X, y ∈ Y, such that every element of X is the first component of exactly one ordered pair in GConsider f=exp (x)1 for a polynomial first prove f (x)=f (a) (xa)g (x) right, sorry, f is a polynomial it was the second part of a question, so they had already mentioned that f is a

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