Suppose that the function f : D → R and g : D → R are uniformly continuous and bounded. Prove that the product fg: D → R is also uniformly continuous. Hint: note that f(u)g(u) – f(v)g(v) = f(u)[g(u) – g(v)] + g(v)[f(u) – f(v)].

Q: Let f : R → R be a function with the following property: |f (a) – f(b)| < 4|a – b| for all a, b E R.

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Q: f= (2xy+ 3x²)ē+ (x² – 2yz + 2y)ē, – (e² + y²)ēz

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

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Q: Prove that there is an additive function T : R --&gt; R that is not linear.

A: Let T: V→W  be the linear transformation and we it is linear,…

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A: To Show: Show that for any function f : X → Y, there are two functions g: S → T and h: X → S such…

Q: Let the function f: R R be defined by f(r) = 3r -5. Prove that the function is bijective.

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Q: A function f: Z x Z is defined as f(m, n) = 3n - 4m.  Verify whether this function is injective and…

A: f: Z x Z is defined as f(m, n) = 3n - 4m So, f(m, n) = f(a, b) implies, 3n - 4m = 3b - 4a i.e. 3(n…

Q: Let f be the function from {a, b, c, d} to {1, 2, 3, 4} with f(a) = 4, f(b) = 2, f(c) = 1, and f(d)…

A: For bijection, the function needs to be one to one and onto.

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Q: Show that f : ℝ → ℝ, f(x) = 5x − 3 is a bijective function.

A: The given function is f:ℝ→ℝ defined by fx=5x-3. We say that a function f is bijective if it is both…

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Q: Show that the function f : R → R given by f(x) = x3 is injective.

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Q: Determine whether the function f : R → R given by f (x) = 4x + 6 is a bijective function.

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