3. Let f, g : [a, b] → [c, d] be cotinuous functions. Suppose that f is surjective. Prove that there exists r E [a, b] such that f(x) = g(x).

Q: Check whether Rolle's theorem conditions are satisfied or not for the given function f(z) =z- 4z on…

A: The given function is: f(x)=x2-4x Now, the conditions of the Rolle's theorem is: 1) f is continuous…

Q: Let f: XY and g: Y→ V be functions. Show the following: (a) f is injective 3h: Y→ X such that hof=…

A:

Q: Let f be a real valued function defined on [a, b] that is differentiable on (a, b). Suppose f′(x) ≤…

A:

Q: et f: AB and g: B→C be functions. Show that if gof is njective, then ƒ is injective.

A:

Q: Show that if f is a meromorphic function on C such that C - ƒ(C) has at least three points then f is…

A: Here's how we can prove that if f is a meromorphic function on C such that C'-f(C) has at least…

Q: Let f: S¹→ S¹ be a continuous function. Proof or disprove the following statements : a) f has a…

A: Given that the  function f : S1 → S1 be a continuous function  where S1 = x,y∈ℝ2 : x2 + y2  =1 We…

Q: Let f : {1, 2, 3} → {a, b, c} be one-one and onto function given by f(1) = a,   f(2) = b and f(3) =…

A: Consider

Q: 3. Let f: A B and g B → A be functions such that gof = i4. Prove that if g is injective, then f is…

A:

Q: Q6. Let f : R? → R² be a function defined by f(t, s) = (1+3t, 2s – t).

A:

Q: Prove that the function f: R → R: x x² + ax + b where a, b E R is continuous at every c E R.

A:

Q: Let f : R → R be defined by f(x) = x^2. Is f one-to-one? Explain your answer.  Is f onto? Explain…

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Q: Let f: X→ Y and g: Y→ V be functions. Show the following: (a) fis injective 3h: Y→ X such that ho f…

A:

Q: Suppose f and g are continuous on a closed interval [a, b] such that f(a) > g(a) and f(b) < g(b).…

A: I have used intermidiate value theoram of continuous function.

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

A:

Q: 3. Let X = {1,2, 3, 4}, Y = {a, b, c} and let f: XY. Prove by contradic- tion that f can't have an…

A: Given that: X={1,2,3,4}and Y={a, b, c}and f:X→Y

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 be a function, and suppose that A is a subset of the domain of f. The image of A under f is…

A: Let f be a function, and suppose that A is a subset of the domain of f. The image of A under f is…

Q: Let I be an interval and f : I → R be a function with Darboux property such that f(I) is finite.…

A: Let I be an interval and f : I → R be a function with Darboux propertysuch that f(I) is finite.…

Q: Consider the function f : R →R defined by f (x) = 2x + 1. This function is:

A:

Q: ) Let f be a continuous function that maps a finite closed interval [a, b] into itself. Prove that f…

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Q: 2. Find functions f,g: (0, 1) -+R with f continuous and g uniformly continuous such that neither fog…

A: Assume any two functions which are uniformly continuous. Now determine the values of f∘g and g∘f.…

Q: Prove or disprove: f : Z30 → Z15 given by f([x]) = [x²] is a well-defined function.

A:

Q: 4. Show that if f:[0,1]→[0,1] is continuous, then f has at least one fixed point. That is, there…

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Q: 4) Consider the function g(x) = [2r – 1] where g : R → Z. a) Is g a one-to-one function? Explain. b)…

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Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.