1. For I = (i1,..., in) € (NU {0})", denote and |I| = ₁ + • + in X¹ = x²¹x²2² A polynomial f : R → R is a function of the form f(x) = Σ αιχ |I|≤d where a, ER is a constant. Show that f is continuous (Hint: write f as a composition of some simple functions that are clearly continuous).

Q: 2. Sketch a graph of a quartic polynomial function y = fx) such that: € fu) > 0 when - 2 1

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Q: Prove that if f:X + Y is a one-to-one and onto function with inverse function f- : Y → X, then fof-l…

A: Given that, f:X→Y is a one-to-one and onto function with inverse function f-1:Y→X

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A: Consider the function Then and The graph of the function is shown below,

Q: Determine the canonical form of the function f : R → R, f(x) = x2 − 4x + 6 and sketch its graph.

A: To determine the canonical form of f(x)=x2-4x+6 and to sketch its graph

Q: Let 4 – x² if x > 1 f(r) if x < 1 Determine whether or not f (x) is continuous at r = 1.

A: If left hand limit and right hand limit same, then f(x) is continuous at x=1.

Q: Derive the autocorrelation function of: x₁ = $xt-1 + εt + 0€t-1 assuming that p < 1, and E(?) = 0²,…

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Q: A1). Sketch the graph of ONE function such that ALL of the following hold: i). f"(x) 0 for x> 2…

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Q: f(x) x² - 2x - 8 { *2-22-3 x-4 x = 4 x‡4

A:

Q: 6. For each of the following, is it a well-defined function? Explain. (a) (b) (d) a: R→Z a(x) = B(n)…

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Q: Let f(x)= 1 Use a 8- e proof to show that f is continuous at x = = 2. X-1

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Q: Let f and g be functions that are continuous on [a, b] and differentiable on (a, b). Show that there…

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Q: Consider the function f with domain (-2, : 3|22 – 1| if – 2 < x < 1 2x2 + 2x – 4 f(x) = 3 if x = 1…

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Q: Suppose n is a natural number, and f: R → Ris (n + 1)- times differentiable. True or false: The…

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