2. Let G be the group of all real-valued functions on the unit interval [0, 1], where we define, for f, g E G, addition by (ƒ + g)(x) = f(x) + g(x) for every x = [0, 1]. If N = {ƒ € G|f(1) = 0}, prove that G/N = real num- bers under +.

Q: The function f : {0, 1}² → {0, 1}³ is defined as: For every x E {0, 1}², ƒ(x) = 0x Select the set…

A: Since f(x)=0x. Thus the range of f is 0(domain).

Q: 3. If the function f : R R is defined by the rule f (x) = 4x + 1 for all real numbers x is…

A:

Q: (b) Given the set S = [-1,1] and the function g: S→ R defined by g(x) = that g is one-to-one, and…

A: g is one to one because  For every x there is a different image in g(x) if x= -1 then it's image is…

Q: 3. Show that f(x) = e describes a bijective function from R to (0, +∞0). (a) Show that f is a…

A:

Q: # 2. Let S be the set of all even integers, and define a function, f: Z –→ S as follows: f(n) = 2n…

A:

Q: Consider the function g : [−3,∞) 7→ R defined by g(x) = (x + 3)2 + 5. Find g −1 by giving a formula…

A: Answer

Q: (a) Prove that, if a and b are nonnegative integers such that a =4 b, then 3ª = 5 3b. ) Using (a),…

A:

Q: Let f: (a) Is f one-to-one? Prove your answer. (b) Is fonto? Prove your answer. (c) Is there a…

A:

Q: Let f be a function defined on an open interval I containing a. Prove that f'(a) exists and is equal…

A:

Q: Consider the function S 2" 13 - 23 if r 1 9(x) Show that g(a) is continuous for all x but NOT…

A:

Q: A function f is continuous on [-3, 5] and contains the points (-3, -4), (0, ƒ(0)), and (5, –3). If ƒ…

A:

Q: 4. Let f: R R to be a contiuous, strictly monotone function. Let f(R) = R. Thus its the inverse f-1…

A:

Q: Define a function f: N → Z by n/2 (1-n)/2 f(n) = { (a) Show that f is one-to-one correspondence. (b)…

A: We need to show that if , then .Let's consider two cases:Case 1:  and  are both even…

Q: Find the limit of f (x) = √(14 + √x), defined on X = [1, ∞)

A: The given function is: f(x)=14+x  is defined on [1,∞).

Q: (1) Let f be a function from A to B, {An} a sequence of subsets of A, and {B} a sequence of subsets…

A:

Q: 4. #x²+x-620, then xs-3 orx 22.

A:

Q: Suppose f and gbare functions with domain R . If both f and g are even but f+g is odd, the prove…

A: Even function :Let us consider  a function f then  f is said to be even if f(x)=f(-x) Odd function:…

Q: ) Prove that, if a and b are nonnegative integers such that a =4 b, then 3ª =5 3b. Using (a),…

A:

Q: (3) Suppose that f: [0, 1] → R is a real function. with f([0, 1]) ≤ [0, 1]. Show that there exists…

A: Sol:- Let g(x) = x - f(x). Then g(x) is also a function from [0, 1] to R. Notice that g(0) = 0 -…

Q: (a) Prove that, if a and b are nonnegative integers such that a = 4 b, then 3ª =5 3b. (b) Using (a),…

A: (a) Let a ≡4 b          => 4 | (a - b)          => a - b = 4k for some integer k.…

Q: k 1. Prove that \k

A: In the first part, it is required to prove that knnk=n−1k−1 We know that nk=n!k!n−k! Now take Left…

Q: (n) If A is a subset of the codomain of f, explain why f(f-1(A)) is a subset of A. (0) Give an…

A: The functions have a property that if x∈Domain of f(x), then, the value of f(x) will be  unique,…

Q: f = 0(g) implies g = (f).

A:

Q: R.2. Can you give an example of a function g:N→N such that g maps N onto N but g is not one-to-one?…

A:

Q: Let f: P(N) P(N) be the function defined by f(X)3{1,2,3,4}. Is f one-to-one? O Yes No What is the…

A: Given: The map f: P(N) P(N) be the function defined by f(X)=X{1,2,3,4}.

Q: a) show that the given function is 1-1 or give a counterexample. b) show that the given function is…

A: (a) Let, f(X) = f(Y) for X, Y ∈ P(N)            => N - X = N - Y           => N - (N - X) = N…

Q: Let Let f: R² R² be the linear transformation defined by be two different bases for R². a. Find the…

A: Given f:ℝ2→ℝ2 be the linear transformation defined by fx→=1-23-3x→   Let B=1, -2, -2, 3, C=-1, -1,…

Q: 3. Let I(r) numbers such that if r # y then g(x) # g(y). Prove that there is a function f such that…

A: