3. Prove that the function f: R – {3} → R - {1} defined by f(x) = , is one-to-one correspondence.

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

A: The solution is given as

Q: f,(x)= e* f2(x)=e¯* (x) = e'

A:

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

A:

Q: Let the function f: R→ R be defined by f(x) = { Question 1 0, x2, x=0, x = 0. (a) Explain the…

A:

Q: Verify that the three conditions of the hypothesis of Rolle's Theorem hold for the function f (x) =…

A: 1. f(x) is continuous on the closed interval [a,b] 2. f(x) is differentiable on the open interval…

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: 4. If f:(X,r)(Y.A) and g:(Y.A) (Z,r)are closed functions, prove that g of :(X.r)(Z.r) is closed…

A:

Q: Prove that the function f : R → R, given by f(x) = 2x, is one-one and onto.

A: Given, function f : R → R, given by f(x) = 2x.

Q: Verify that the three conditions of the Rolle Theorem hypothesis hold for the function f (x) = 1-2x…

A: Three conditions of the Rolle's theorem hypothesis. Consider function f(x), it should be continuous…

Q: 1. Find and sketch the domain of the function f(x, y)= In ツー1 Vアーx

A: As per our company guidelines we are supposed to answer only one question at a time. Kindly repost…

Q: Assume that f is an odd function defined on R and t e Rt. Compute

A:

Q: Show that the function f : R → R,defined as f(x) = x2 , is neither one-one nor onto.

A: We will show this by counter examples.Let x= 2 and -2 Then f(2)= 2^2 = 4 and f(-2)= (-2)^2 = 4 This…

Q: The function f is defined by f:R→ R/{-1} , f:x*-! x+1 **-1 and hf (x)= 2x – 3. (a) Find h(x). (b)…

A:

Q: Let f, g: R → R defined by if x 1 f(x) = %3D and g(x) = 3x + 2. %3D fog and go f.

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Q: Let f: R (-1, 1) be a function defined by Prove that (a) f is injective; (b) f is surjective. f(x) =…

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Q: Prove that the function g: R > R defined by f(x) = 1+2^x is one-to-one but not onto.

A: Consider the function f:R->R defined by:

Q: Prove that the function f: R –→ R defined by f (x) = 3x – 5 is bijective.

A:

Q: Let f, g:R R defined by if x 1 (x2 and g(x) = 3x + 2. %3D Find f-1 and g-1.

A:

Q: Prove that the function f: R -→ R defined by f(x)= x³ is bijective.

A: If a function is both one-one and onto, then the function is bijective.

Q: Show that the function f : N → N, given by f(x) = 2x, is one-one but not onto.

A: If f(x1) ≠ f(x2) then we get:

Q: Determine the c which satisfy the conclusion of Rolle’s Theorem for the given function and interval.…

A:

Q: 1. Consider the functions f : A → B and g : B → C. (a) If f and g are both surjective, show that so…

A:

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

A: As you post multiple questions According to guidelines I only solve the first one.

Q: Let f be a function defined on the real line R and suppose that it satisfies the condition If(x) –…

A:

Q: Consider the function f : R³ → R defined as V(x1, x2, x3) E R³ f(x1, x2, X3) = X1X2+ X1X3 + X2X3 a.…

A: We are given the function f: ℝ3→ℝ defined as ∀x1, x2, x3∈ℝ3    fx1, x2, x3=x1x2+x1x3+x2x3. For (a),…

Q: Let f:Z → Z be defined by f (x) = 2x + 1 where x € Z. Is f a one to one correspondence (bijection)?

A: Now let

Q: Show that the function f : R→ R : f(x) = x° is one-one and onto.

A:

Q: 5. (a) Consider the function f : C → R, f(2) = (z + z)². Write down the %3D domain, codomain and…

A: The domain is C, the set of all complex numbers. The codomain is R, the set of all real numbers.

Q: If f is a function defined from R to R, is given by f(x) = 5x - 3 then f -(x) is given by %3D -- 20…

A:

Q: 3. Consider the function Prove that f'(0) = 0. & f(x) = if r EQ if x & Q.

A: Introduction: A differentiable characteristic of 1 actual variable is one whose spinoff takes place…

Q: 2. Let g : R → R be a bounded function and define f : R\ {0} → R by 1 f(x) = g(x)+ x2 Using only the…

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Q: f(2) = e=².

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Q: Let f : R → R be continuous such that f(q) = 0 for all q ∈ Q. Prove that f must be the zero function

A:

Q: Let f (x, y) = 4-y² In(y–z²) %3D Draw a picture of the domain of this function.

A: x,y ∈R2

Q: Theorem 8.9. Let f : X Y be a continuous, surjective function. If X is connected, then Y is…

A:

Q: 7. Show that the function f : R → R* defined by f(x) is not onto. x2 +1

A:

Q: Prove that the function f: R W {2} R {5} defined by - 5x + 1 f(x) = #-2 is bijective.

A:

Q: Let h(x) = cos x and g(x) = x - l. Compute the composite functions h o g and g o h, and find their…

A:

Q: -B-Let f:( M, (Z).+-)-(M, (Z). +..) be a function define by such that g[]=[ EM₂(2). Is f is…

A:

Q: Suppose the function f has the property that there exists a number B such that |S(x) – f(c)| < B|x –…

A: Suppose the function f has the property that there exists a number B such that |f(x)-f(c)|≤B|x – c|…

Q: 6 Suppose that a function h satisfies –x² < h(x) < x² for all x. Prove that h is differen- tiable at…

A: Definition used: f'x=limh→0fx+h-fxh and f'a=limh→0fa+h-fah Calculation: Consider the given…

Q: If f(z) u * 1V is an analytic function of z then prove that 2B. Əx?

A:

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: Let the function f: R→ R be defined by { f(x) = 0, 1 x2, x=0, x = 0.

A:

Q: Form the PDE by eliminating the arbitrary functions from the relation z = f(xy)

A:

Q: 5. Find all analytic functions f such that Ref + Im f = xy.

A: To find all analytic functions f such that Re f + Im f=xy.

Q: 9. Show that if the analytic function f(z) has a zero of order N at zo, then f(z) satisfying g'(zo)…

A:

Q: If f is a function of only r, g is a function of only y, and f and g satisfy In(f(x)) + In(g(y)) =…

A: Given: f is a function of the only x and g is a function of the only y.…

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.