Let A and B be subsets of a set X. Prove that the functions f(x) = XAUB(x) and g(x) = XA(x) + XB(x) – XA(x) · XB(x) are equal.

Q: Suppose f and g are non-constant, differentiable, realvalued functions defined on (−∞, ∞).…

A: We plug x=0 and y=0

Q: Let f(x) =(x-1)(x-2)(x-3),Prove that there is more than one "c" in (1,3) such that f'(x) =0

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Q: Let f(x) = 2x and g(x)= 2x2, and p; = {0, 1, c} with c > 1• a partition of the interval [0,c)…

A: In the given question we have to find the value of c .wheref(x) =2x and g(x) =2x2 and given that the…

Q: Prove that f(x) = x 3 - 12x is differentiable. Compute f'(x) and find f '(-3), f '(0), f' (2), and…

A: Since, we know  that a polynomial function is always differentiable.⇒f(x)=x3-12x-(i) is always…

Q: Let f: A→ X X Y be defined as f (a) = (f₁ (a). f2 (a)) each a ≤ A, Then f is continuous if and only…

A:

Q: Find f g and g of, where f (x) = x^2 + 1 and g(x) =3x + 2, are functions from R to R.

A:

Q: Let f(x) = m1x + a and g(x) = m2x + b, where m1, m2, a, and b are constants. Determine the necessary…

A: f∘g=g∘ffgx=gfxfm2x+b=gm1x+am1m2x+b+a=m2m1x+a+bm1m2x+m1b+a=m1m2x+m2a+bm1b-b=m2a-abm1-1=am2-1

Q: 6. Let A and x be defined as A = X = and define the function f(x) = x" Ax. (i) Show that Vf(x) =…

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Q: 1. Let f be defined on an interval I and let x1 and x2 be numbers on I. If f(x1) < f(x2) whenever x1…

A: Given that: f is deifned in I such that:   f(x1)&lt;f(x2) whenever x1&lt;x2

Q: Verify if the function f(x)= x² – x – 30 satisfies criteria stated in Rolle's theorem on [-5, 6 ]…

A:

Q: If f(a) > f(x) in an open interval containing a, x + a, then the function value f(a) is a relative .…

A: Given: If f(a)&gt;f(x) in an open interval containing a, x≠a, then the function value f(a) is a…

Q: If f(a) &gt; f(x) in an open interval containing a, x ≠ a, then the function value f(a) is a…

A: Definition:

Q: Exercise 1. Let f : A → B be a function with X a subset of A. Prove that if f(X) = 0, then X = 0.

A:

Q: If g(x) = x' In(x), for x> 0, then determine g'(x

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Q: A function f is ___________ on an interval I if, for any choice of x1 and x2 in I, with x1 &lt; x2,…

A: To fill: The blank of the statement. Given: The statement is "A function f is _______ on an open…

Q: For any subset A of R, define the function XA : R →R by if x E A 1, XA(") = 0, if x ¢ A. Let A and B…

A:

Q: Let f(x) = x3. Verify the mean value theorem by finding a c (−2, 1) such that f '(c) =  f(1) −…

A: f(x) = x3f(-2) = -8; f(1) = 1f'(x) = 3x2f'(c) = 3c2

Q: Let f(x)=x−2x+2. Find all values of c that satisfy the Mean Value Theorem for the function on the…

A: We seek to verify the mean value theorem for the function on the interval [-1,2]. According to the…

Q: For any subset A of R, define the function XA : R → R by [1, XA(x) = 0, if π A if x ¢ A. Show that…

A:

Q: Let f: X →Y be an injective function, and A, B C X. Prove that Q1) f(A)n f(B) C f(AN B).

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Q: Suppose that f : A → B and g : B → C are both injective functions, prove that the function go f : A…

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Q: Let f(x) = x2 – 3x. Use Rolle's Theorem to find all the values of x = c in the interval [0, 3] such…

A: Use rolle's theorem As per guidelines only first question solve raise another queu

Q: Find a function f and a real number a such that

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Q: f(x) = {*- Įx² +px + 2,x # 3 3, x = 3

A:

Q: A function f is ________ when there exists a positive real number c such that f (t + c) = f(t)for…

A: We know that for a functionft, if there exist a positive real number c such that ft+c=ft, for all t…

Q: Let ƒ(x) = px2 + qx + r be a quadratic function defined on a closed interval [a, b] . Show that…

A: Given,

Q: Let f : A → B be a function. Prove that f is surjective if and only if, for every pair of functions…

A: Please comment if you need any clarification. IF you find my answer useful put thumbs up. Thank you.

Q: 6. Let A and x be defined as A = X%3D %3D 9. and define the function f(x) = x" Ax. %3D (i) Show that…

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Q: Let f : A → B be a function. Prove that if C C A and D C A and f is injective, then f(C) = 7 (D) =→…

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Q: Determine the values of m and b such that function g is continuous at every real number: if r 1.

A: Since the function g is continuous.So, limx→1-g(x)=g(1)limx→1-x2-mx=-212-m(1)=-2m=1+2m=3

Q: Determine the global extreme values of the function on the given set without using calculus. f(x, y)…

A:

Q: For a function g(x), if there exists x E [a,b] such that g'(x) > 1, then we conclude that g(x) has…

A: Solution: Given, gx is a function and there exists x∈a,b such that g'x&gt;1. Then, gx has not a…

Q: Let g : Z x Z -→ Z x Z be the function given by g(x, y) = (3x + y, x – y). Find, with proof, an…

A:

Q: Check the validity of the Rolle’s theorem for the functionf(x)=x2−4x+3x−2on the segment [1,3].

A: Given:

Q: Let f, g : R→R be defined, respectively by f(x) = x + 1, g(x) = 2x – 3. Findf + g, f – g and f/g.

A:

Q: Let f(x) = x² – 5x. Using Rolle's Theorem, what are all the values c in the interval [0, 5] such…

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Q: Let A be a nonempty set and let f : A → B be a function. Prove that f is one-to-one if and only if…

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Q: Prove that for any sets C,D⊆Y and for any function f:X⟶Y, f−1(C∩D)=f−1(C)∩f−1(D).

A: We will prove the given statement.

Q: Let f(x) = x2 and let P = {-2, -3/2, -1, -1/2, 0, 1/2, 1 }. Compute Lf (P) and Uf (P). Lf (P) =  Uf…

A: Given, fx=x2P=-2,-32,-1,-12,0,12,1 Formulas used: LfP=∑i=1nmixi+1-xiUfP=∑i=1nMixi+1-xiWheremi=inf…

Q: If f(a)&gt; f(x) in an open interval containing a, x ≠ a, then the function value f(a) is a…

A: If f(a)&gt; f(x) in an open interval containing a, x ≠ a, then the function value f(a) is a relative…

Q: Prove that the function is odd. f(x) = a2n+1x2n+1 + . . . + a3x3 + a1x

A: Consider a function: fx=a2n+1x2n+1+a2n-1x2n-1+⋯+a3x3+a1x

Q: Let [a,b] be a closed and bounded interval, let keR, and let f:[a,b]¬R be a function. Suppose that f…

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Q: Let f be a function on an interval I and let ₁ and 2 be any two points in I. Then if f (x₁) > f (x₂)…

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Q: If f and g are continuous functions on the closed interval [a, b], a, b e R and g(x) > 0 for all x…

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Q: Let g : R → R be the function g(t) = t, and let B = [-3, 1] U [2, 3]. Express the set g(B) as an…

A: First observe the behavior of the given function  g(t)=t2 : Domain of the function is ℝ and range…

Q: If a function f(x) = x(x -2) satisfies conditions of Rolle's theorem on [0,2] , then the value of…

A: To find the value of c

Q: Let g be defined on an interval A, and let c ∈ A. (a) Explain why g'(c) in Definition…

A: Given that function g is gefined on interval A and c∈A

Q: Let f : A → B be a function between sets A and B. Let X1, X2 ⊆ A and Y1, Y2 ⊆ B. a) Prove that if…

A: 1. A collection of well-defined objects or elements is called a set. If a is an element of a set A,…

Q: Verify if the function f(x)= x² – x – 30 satisfies criteria stated in Rolle's theorem on [-5, 6 ]…

A: