- (i) Determine whether the function f: R - R, f(x) = X+1 x*3 is surjective. x-3"

Q: В. B. Show that the function f (z) = -i z2 is analytic.

A: The solution is given as

Q: 2. Prove that the function f : R? → IR defined by f(x, y) = 1- a2 – 4y? is neither injective nor…

A: Given the function f:ℝ2→ℝ defined by fx,y=1-x2-4y2

Q: 7- The function f(z) = z is: a) Analytic on C b) Analytic only at zo = 0 c) Not analytic

A:

Q: b) Let f be a differentiable function of one variable, and let z = f(3x² – 5y²). Show that az əz…

A: Given  z = 3x2 - 5y2

Q: Find the antiderivative F of f(x)=4-3(1+x^2)^-1 that satisfies F(1)=-2.

A:

Q: : Apply Rolle's theorem for the function f(z) = 7x² – 28z+ 21 over the interval [1,3]. er.

A: The given function is, f(x)=7x2-28x+21 over the interval [1,3]. Observe that, the given function…

Q: → [0, ∞) defined by f (x) = Vx + 2. Prove that f is 17. Consider the function f: (-2, 0) surjective.

A:

Q: Q3/ 1- Let f: Zn → Z10 define by f(x) = 2x+2. Is fhomomorphism or not, explain your answer.

A: Given :-  let f : ℤn → ℤ10 define by f x = 2x + 2.  To Determine :-  Is f homomorphism or not.

Q: Determine whether the function f: R- R, f(x) =,x # 3 is surjective. x+1 %3D r-3

A: Consider the provided question, We need to determine whether the function f:R→R fx=x+1x-3, x≠3 is…

Q: Use the Second Fundamental Theorem of Calculus to find F′(x).

A: Given:          F(x)=∫−2x(t2−2t)dt

Q: Discuss the continuity of the function. f(x, y, z) = 1/(√x2 + y2 + z2)

A: Given rational function is

Q: Define the function f(x, v) = 4x³-x³y-4y5 3x5-3y5 at the origin so that it becomes continuous there.

A: A function is said to be continuous at (a,b) if the the value of function at (a,b) is equal to…

Q: 5. Prove that the function f: Z → Z defined by f(x) = 3x + 1 is one-to-one.

A:

Q: f(x)={x^2−9x≤c, 2x−10 x>c } i

A: Consider function f(x)=x2-9           x≤c2x-10        x>c Now  limx→c+ f(x)=limx→c+…

Q: Explain why f (x) = sin (x² + 3r + 1) is a continuous function for all c € R. %3D

A: Let us assume that: g(x)=sin(x)and h(x)=(x2+3x+1)So,…

Q: D. Discuss the continuity of the function g (x): 1 %3D on the interval [-1,2]. x2-4

A: Given, To discuss continuity of the function g(x)

Q: Let I be an interval disjointed from {-1, 1} . Prove that the function f(x) = x + (1/x) is…

A:

Q: 10. ( ) Find the absolute extrema of the funcion f(x) = 3 Vx2 - x on the interval -1, 1.

A:

Q: Let f(æ) = – 3x* In æ %3D f'(x) = f'(e²) =

A:

Q: Let f(x)=x4 + 1/3 x3 - 7x2 Identify the interval(s) where f is concave upward.

A: The given function is: f(x)=x4+13x3-7x2 To find the concavity, find the second order derivative of…

Q: Let f(x) = x -[r] and f :[1,5) →[0,1). What is the discontinuously type of f(x) at the point x = 3?

A:

Q: Use the Second Fundamental Theorem of Calculus to find F′(x).

A:

Q: Let f : R → R with f(x) = 3x² + 2. Prove that f is differentiable at the point –1 %3D

A: f(x) =3x2+2 f'(x) =limh→0f(-1+h) -f(-1)h        =limh→03(-1+h)2+2-3-2h        =limh→03+3h2-6h -3h…

Q: Find and sketch the domain of the function f(x,y)= exy/ x-y2

A: See the attachment

Q: Consider a function f(x) = |x – 3|. a. Show that f (x) is not differentiable at 3. -

A: Given         f(x) =|x-3|

Q: Can Rolle's Theorem be applied to f(x) = 1/(1-x) on interval [0,3]? If yes, what is c in (0,3) such…

A: here we can't apply the rolles Theorem

Q: 20. Express the function F(x) = 1/x + J three functions. X as a composition of

A:

Q: 2. Determine if a function f : [0, 27] → R defined by f(x) = sin 2n is continuous.

A:

Q: Determine whether the function 2х + 1 f(x) %3D 2r2 2x2 – x – 1 is continuous: a. at 2; b. at 1.

A:

Q: Q2. Show that F(x) = 3 – x², x € is contractive on its domain, but does not have any fixed points.…

A: To show - Show that Fx = 3 - x2, x ∈ -14, 14 is contractive on its domain, but does not have any…

Q: f(2) = e=².

A:

Q: what is the antiderivative F(x) of f(X) = 3x1/2-x-1 that satisfies the condition  F(4)= 0? A,…

A: The given function is fx=3x12-x-1 that satisfies the condition F4=0. Find the antiderivative of Fx.

Q: Discuss the continuity of f(x) = {Vx, x > 1 1, x<1 at x = 1.

A: A function f is said to be continuous at some point a if the value of the function at that point is…

Q: 4. Give an example of a continuous function f : (0, 1) → R that cannot be uniformly approximated by…

A:

Q: The value of c of f(x) = 2-, in the interval (1,3) such that f'(c) = %3D f(3)-f(1) %3D is 3-1 V340…

A: Given fx=2-3x given interval 1,3 f'c=f3-f13-1

Q: Determine the maximum value of f (x) = 5x – e5x [T:3] on the interval [-1,1] %3D

A:

Q: Determine the set of points at which the function is continuous. F(x, y) - х-у 1 + x² + y2

A:

Q: 2. Use the First Fundamental Theorem of Calculus to differentiate the function -5x f(1) = | V1+ tª…

A: To differentiate the function by using first fundamental theorem of calculus f(x)=integration (x to…

Q: A continuous real-valued function f (x) with f (a) ƒ (b) < 0 , has at least one root in [a, b]

A: This question is related to Real Analysis.

Q: O Show that the function f: N2 N defined by f(x, y) = 2X3Y is injective but not surjective.

A:

Q: Is the function f:[0.1]→R: f(x) =- -(2x+3}** at x=0.1 9x+2 a) continuous and differentiable b)…

A: The given function is: f:0,1→ℝ:fx=-2x+3509x+2 at x=0.1

Q: 1. Explain what it intuitively means for function f(x,y) to be continuous in some domain D.

A: To explain the meaning of the given statement.

Q: Q1// Find and sketch the Domain of f(x,y) In (x' y)

A:

Q: 3. Show that there does not exist a non-constant continuous function f : R2 → R such that f(x, y) =…

A: Given:- f: R2→R

Q: A function F is an antiderivative of the function f if F'(x) = %3D for all x in the domain of f.

A: We have to solve given Questions.

Q: Let I be an interval disjointed from (-1,1) Prove that the function f(x) = x + (1/x) is increasing…

A: Solution: Consider the given function is: f(x) = x + 1x    ...(1) Also, the given interval Let I is…

Q: 16. Consider the function f:(-2, 0) → [0, 0) defined by f (x) = vx +2. Prove that f is injective.

A: f(x) here is injective if and only if for all a,b∈(-2, ∞) we have a=b⇔f(a)=f(b)

Q: Prove that f(x) = x 3 - l2x is differentiable. Compute f'(x) and find f '(-3), f '(0), f' (2), and…

A: Given information: The function is f(x)=x3-x. Calculation: We need to find the value of f'(x) and…

Q: 6x2 - x²y +6y2 x2 + y2 Define f(0, 0) in a way that extends f(x, y) = to be continuous at the…

A: The given function is- fx,y=lnx2y-x Partially differentiating it w.r.t. x- ∂f∂x=2xy-1x2y-x Again…