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 interval, and provide a proof for your answer.

Q: f,(x)= e* f2(x)=e¯* (x) = e'

A:

Q: Prove that there does not exist any function f(x, y) such that

A: To show that there does not exist any function that satisfies the given conditions

Q: Find the Wronskian for the set of functions.{x, e−x, ex}

A: Find the Wronskian for the set of functions.

Q: (b) A continuous function f : [0, 1] → R such that its graph (r, f (x)) C R² is not closed.

A: The given statement, "A continuous function f:0,1→R such that its graph x,fx⊂R2 is not closed" is…

Q: Prove that if f is a continuous function on a closed interval [a, b], then

A: We have to prove the given inequality for a continuous function on a closed interval [a,b].

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 be a real valued function defined on [a, b] . Let f have a local minimum at a point x ∈ (a, b)…

A:

Q: Without the aid of the Wronskian determine whether the given set of functions is linearly…

A: Given functions: f1x=lnx and f2x=lnx2 To show: Determine whether f1x and f2x are linearly dependent…

Q: e*, cos x, sin x

A:

Q: (b) Show that the u(x, t) can be written as u(x, t) = 0(x – at) + 4(x + at), where o and are…

A: The multivariable function u(x,t) is a rule that associates or fixes a number to a particular point…

Q: The domain of the function f (x, y, z) = In(6 - 3xy) %3D is {(x, y) E R? : xy 2} {(x, y, z) E R :…

A:

Q: Suppose that f'(x) > M for all x E [a, b]. Prove that f(b) > f(a) + M(b

A: Since  f'(x)≥M therefore the function is differentiable and hence continuous let f is continuous on…

Q: prove that any Continuouse function [a,b] en is measurable.

A:

Q: t, 0<t<1 f(t) -1, 1, 1<t< 2 || –t, 2<t<10

A:

Q: Let f : [a, b] R be a continuous function on [a, b). Prove that if |f(x)| dx = 0 then f(r) = 0 for…

A:

Q: Let f : R² → R² be a continuous function. Prove that f([0, 1] × [0, 1]) is compact.

A: It is given that f:ℝ2→ℝ2 be a continuous function. Prove that f0, 1×0, 1 is a compact set. Let Gi:…

Q: Let A > 0 be a fixed but unspecified constant, and let f(x) = xn= (In x)", x > 1. Compute f'(x).

A: our objective is to find the derivative

Q: Let g : R → R be a positive function such that g(0) = 1 and g(x+y) = g(x)g(y), ∀x, y ∈ R. Show that…

A:

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: fi (x) = x, f2(x) = cos x, fa(x) = 5

A:

Q: f (x)dx = f(c)(b – a). a

A: given: the function f is continuous on the closed interval [a, b], then there exists at least one…

Q: Let f and g be continuous functions on [a, b], and suppose that f(a) g(b). Prove that f(c) = alc)…

A:

Q: Let g(x) = 8 In(t) dt. Use the Fundamental Theorem of Calculus to find g'(x). gʻ(x) =||

A: The given function,         gx=∫1x8 ln(t) dt

Q: Prove whether u(x, y) = x² – 2xy + y² is a harmonic function on a set D= {(x,y) | x < y}. I

A: Here, the given function is ux,y=x2-2xy+y2 on a set D=x,y|x&lt;y. The function is a harmonic…

Q: Let f:Z x Z+ Z x Z be the function given by f(z, y)= (2z + y,I- y). Find, with proof, an element af…

A:

Q: Let f(2)=u+iv be an analytic function. Given that u=v² show that f(z) is a constant.

A:

Q: Let T be function from R to R2 de fined as T (x) = (0, x), Then show that if T is linear operator or…

A:

Q: Prove that if f is a real-valued continuous function on [a, b] × [c, d], then f takes on a maximum…

A:

Q: 1 The domain of the function f(x,y) is the set of all ordered pairs (x , y) such that : Oy + +x Oy -…

A: We have a given function F(x,y) .and finding the domain of function

Q: Consider a function space of the set of functions f(x) of a real variable x ∈ (−∞, +∞), such that…

A:

Q: Let f : [0, 1] → R be a function with Darboux’s property such that for any y ∈ R, the set  f…

A:

Q: Let X and Y be t function discrete ran X--1 1, 2. y-4, 9 27 P(x, y)- 0.

A: Two dimensional random variable is said to be discrete when it takes at most a finite number of…

Q: ) Let f be a continuous (and thus uniformly continuous) function on [a, b]. Show that, given e > 0,…

A:

Q: problem in photo attached

A: The given function is,

Q: Let f : (0, 1) → R given by f(x) = 1/x . (i) Prove that f is continuous. (ii) Prove that f is not…

A:

Q: Let A := (0,1] and let f : A → R be defined by f(x) = . Prove that f is continuous on A.

A: Consider the provided question,

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

A: Take a point (1, 0) and suppose it belongs to the range set.

Q: The domain of the function f (x, y, z) = In(6 – 3xy) %3D is {(x, y) E R² : xy 2} {(x, y, 2) E R³ :…

A:

Q: Consider the function f : [a, b] → (a, b] with ƒ being continuous. Show that there is a number c in…

A:

Q: Let T be function from R to R² de fined as T (x) = (x, 0), Then show that if T is linear operator or…

A:

Q: Let f:C → C be an entire function. Show that the function g:C \ {0} → C defined by g(z) = f(1/7) is…

A:

Q: Find the Wronskians of the given sets of functions and determine whether the functions are linearly…

A:

Q: B. Show that the space V = {f : R → R} of functions from R to R is not finite- dimensional.

A: Let V=f:ℝ→ℝ. Let A is set of all polynomial with integer coefficient. We know that every polynomial…

Q: Let f be a C' function on the line, and let g(x) = S f(xy)y² dy. Prove that g is a C' function and…

A: f ∈ Cn  means f is atmost n times differentiable

Q: Consider the Cobb-Douglas function n f (X1,x2,..,Xn) = x"x. i=1 where x; > 0, and a¡ > 0 for all i=…

A: Consider the Cobb-Douglas function fx1,x2, ... , xn=x1α1x2α2x3α3... xnαn=∏i=1nxiαi Find the…

Q: Consider the function f : Z × Z → Z × Z defined by f(a, b) = (a+ b, a b) -

A: