6. The function of f: R → R, g: R → R are defined by; S1 – x, if x > 0 f(x) = {;?, if x < 0 Sx, g(x) = {x- 1, if x > 0 if x < 0 (a) Find a formula for f • g. Show that f o g is bijective and find its inverse. (b) Find a formula for g •f. Show that g of is neither injective nor surjective.

Q: 5. Show that the function h(x, y) defined by if r2 + y? 1 is discontinuous.

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Q: 1. Show that for all f e L'(R) lim dr = 0. %3D

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Q: (a) Find if y = - (In z)*". (b) The function f(x) = (In z)* is an invertible function (you do not…

A: Given query is to find the derivative of the expression.

Q: Q2. Let f:R- R be a function such that f(x) = x sin| f is defined at x, , where the value of d…

A: Topic- Limits

Q: 8. Given the following function f, define f(1) so that f is con- tinuous at x = 1, if possible: Зх —…

A: Given: fx=3x-1if x&lt;1x2+1if x&gt;1

Q: 1) Find the value of C that make the function continuous at x-1? x 1 -

A: We have given piecewise function  Function is continuous at x=1 if  f1+=f1=f1- f(x) = C-x        if…

Q: 2. (a) Suppose that f is a differentiable function such that |f'(x)| < c for every x. Show that for…

A: To Prove: a. If f is a differentiable function such that f'(x)≤c for every x.Then for any real…

Q: 16. Suppose that f is differentiable at c and that f'(c) + 0. Show that there exists a ð > 0 such…

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Q: 10, Let f be the function defined by the following. sin x, x<0 x. Osx<1 S(x)={ 2-x, x-3, x22 For…

A: Given: the function f defined by the following             fx=sin…

Q: 2. Which of the following functions satisfy the hypothesis of Rolle's Theorem on the interval [0,…

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Q: 2. Suppose f20, f is continuous on [a, b], and f(x) dx = 0. Prove that f(x) =0 for all x e [a, b).…

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Q: Let f(x) = e x if −π < x 6 π. And extend f to a 2π periodic function on R. Determine the…

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Q: 1. Suppose f is a function which satisfies f(x + y) = f(x) + f (y) for all x,y e R. What is the…

A: Let f is a function which satisfies f(x+y) = f(x) + f(y)  for all x,y∈R   f(x+y) = f(x) + f(y) ,   ∀…

Q: The antiderivative of f(x), denoted by F(x), exhibits an odd symmetry i.e., it satisfies the…

A:

Q: (Q4) Prove directly from the definition that the function f: R→ R defined by f(x) = 3r² +r for all r…

A: Sol

Q: 4 Given that f(2) is analytic at z = 0 and that f G) =, n = 1,2, 3..,find f(2). VE

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Q: 3. Consider the following formula f (x) = lim n-0 gn +1 Let D = {x € R: f(x) E R}. Calculate f(x)…

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Q: 1. Let the function f : R → R be given by 1- cos(x) if x + 0, f(x) := if x = 0. 1 Prove that f is…

A: Given : f'(x)=1−cosxx  for x≠0f'(x)=0    for x=0

Q: f(x + 2h) – 2f (x+ h) + f(x) lim h→0 6(2.x + 1) || h2

A: Consider the given limit limh→0fx+2h-2fx+h+fxh2=62x+1 Then,…

Q: 2. Consider the function if x = in lowest terms q > 0, p, q € Q; f(x) = O if x e Q°,x = 0 Question:…

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Q: 1 Let f (x) = ; 1 , then f (x) satisfies the - - 2 conclusion of Rolle's Theorem on 0, 1| 6.

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Q: 15.1. Using the results of the preceding exercise, show that the function g, defined on R to R by…

A:

Q: If we apply Rolle's Theorem to the function f(x) = 2x2 – 16x – 4 on the interval [2, 6], how many…

A: We state Rolle's theorem here ------    Let f : [a,b] -&gt; R be a function continuous on [a,b] and…

Q: 10. Consider the function F: R² → R defined as: { In (1 + y²) – y² (ev + 4.x) if y +0 F(x, y) = yA…

A: Solution is in next step

Q: Find the absolute maximum and absolute minimum of f(x) on the given interval f(x) = 2x2 – 4.x +3,…

A: To find the absolute minimum and maximum of a continous function on the closed interval [ a, b ], we…

Q: 1. Employ the following methods to find the maximum of f(x) = 4x – 1.8x² + 1.2x – 0.3x* (a)…

A: Given f(x)=4x-1.8x2+1.2x3-0.3x4. To find: (a) find the maximum using golden search method.…

Q: , 6]. che function on this interval. 1/sqrtx ow there exists a c in the open interval (2, 6) such tl

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Q: X if x e Q If f (x) is defined on [0,1] by the rule f (x) = < (1-x , if x 4 Q Prove that fof (x) = x…

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Q: 1. Apply the Parseval's identity to the function f (x) = x, –n < x < n

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Q: Suppose that f: R → R and g : R → R are related by the formula f (x) = g(x)+(x-a)". For which values…

A: Note: A function fx is tangent to the function gx at a point x=a if (1) fa=ga, and (2) f'a=g'a

Q: 16.C. If f is defined and continuous on R? to R and if f(xo) > 0, then is f positive on some…

A: Consider the provided question,   We need to show that, f is strictly positive on some neighborhood…

Q: Does the Rolle's Theorem applies for f(z) – ln(2 – z²) over the interval [−1, 1]? O Cannot be…

A: This is a problem related to derivative of the function. Based on the general formula and concept we…

Q: 3. Consider the function f: ICR → R that is continuous and strictly increasing on I. Then the…

A: True.

Q: Q2. Let f:R→ R be a function such that f(x) = x sin (-) has a limit at xo + 0. If f is defined at xo…

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Q: 10.C. If f is defined and continuous on R to R and if f(xo) > 0, then is f positive on some…

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Q: 5. Show that the function h(x, y) defined by 1 if a? + y? 1 is discontinuous.

A: We can solve this using R.H.L ≠ R.H.L

Q: Find the value of a such that the function S(«) = {** -1, x < 3 x 2 3 | f(x) = 2ах, is continuous at…

A: Given:      f(x) = x2 - 1  , x &lt; 3     f(x) = 2ax  ,   x ≥ 3

Q: Consider the following equation defining the function f on the integers implicitly: f(n) = n+ f(n –…

A: Given:            f(n) = n +f(n-1),  f(1) = 1

Q: 10. Consider the function if a E Q -a? if x ¢ Q. f(x) = Prove that f'(0) = 0. %3D

A: Solution:Given  f(x)=x2if x∈Q-x2if x∉QTo prove f'(0)=0Formuladdx(xn)=nxn-1

Q: Let f be the function defined on [0, 1] by when x is irrational, f (x) = 1 when x is rational.…

A: Please see the below picture for detailed solution.

Q: Find the antiderivative  F of f that satisfies the givencondition. Check your answer by comparing…

A: Given: f(x)=x+2sinxF(0)=−6

Q: 1. Suppose that f(z) is analytic for lz <1 and sa Show that there is a constant c < 1 such that…

A: To prove the estimate for }f(z)| over the disc or radius r, center 0

Q: (1- a? f(x) = 1– x² if x E Q, if r ¢ Q. %3D p at which f is continuous. Prove your conclusion.

A:

Q: 1. Consider the function z = f(x, y). If fa(a, b) and fy(a, b) both exist, and they are equal, then…

A: 1. Given: z=fx,y and suppose that fxa,b and fya,b both exists and they are equal and it is…

Q: 15. If f+g is differentiable at a, are f and g necessarily differentiable at a ? If f g and f are…

A: Let f be any function that is not differentiable  at the point a and define as: gx=-fx for all real…

Q: The antiderivative of f(x), denoted by F(x), exhibits an odd symmetry i.e., it satisfies the…

A:

Q: A function f is defined by f (x1) = / Cos(x – t) os(x-t) dt cos(t) f (x) : 01 Simplify f(x) and…

A:

Q: 3. Consider the function g(x) defined by { x² sin(1/x) if x > 0, g(x) = if x <0. (a) Show that g(x)…

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Q: 2. Let f(r)= r on (0,1]. Show that fe R(0, 1] and find f f. %3D

A: