Let f and g be functions which are differentiable on R. For each of the following statements, determine if it is true or false. If it is true, then prove it. If it is false, then give a counterexample (you must prove that it is indeed a counterexample). (a) If f'(x) = g'(x) for all x, then f(0) = g(0). True False (b) If f'(x) > sin(x) + 2 for all x € R, then there is no solution to the equation ef(x) = 1. True False

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A: The given graphs are of f(x) and g(x). Here, note that the graph of a differentiable function has a…

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Q: Let X={1, 2, …, 16} and define function f(x)=rx for all x€X where rx is the remainder when 3x is…

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Q: Let f(x) = m1x + a and g(x) = m2x + b, where m1, m2, a, and b are constants. Determine the necessary…

A: f∘g=g∘ffgx=gfxfm2x+b=gm1x+am1m2x+b+a=m2m1x+a+bm1m2x+m1b+a=m1m2x+m2a+bm1b-b=m2a-abm1-1=am2-1

Q: Prove formally that for all a > 0, function af (x) + g(x) is also a concave function.

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Q: Suppose f and g are positive functions. Is the following true or false? Vf,g:N → R f(x) E O(g(x)) =…

A: Big-Θ notation: A function g(n) is in Θ(f(n)) if there exist constants c1, c2, and N such that…

Q: Explain why there does not exist differentiable functions f and g for which f(0) = g(0) = 0 and =…

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Q: Let f(x)= 2/x-3. Let g(x)= 5/x+2. Determine (f/g)(x)

A: See the attached file

Q: Determine F(x) if, for all real x and y, F(x)F(y) – F(xy) = x + y.   Hint:  What could F(1) be?

A: Consider the provided question, We have to find F(x).

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Q: Consider the function f(x) = -x + b, where b is a constant, and find the following. a. f(f(0)) b.…

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Q: The table contains some input-output pairs for the functions f and g. æ| f(x) | g(x) 2 2 9 6 9 3 9.…

A: To evaluate the given expression with the given table values.

Q: 15 10 8 10 Evaluate f(0) (a) Estimate f(4) (b) 00 2.

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Q: Let ƒ(x) = x/x - 2. Find a function y = g(x) so that (ƒ ∘ g)(x) = x.

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Q: Let f(x) = ∥x∥. Find out its conjugate function f ⋆ and prove your conclusion.

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Q: Let f(x) be the function x + 5 for x 2.

A: Let f(x)=x+5 for x<-1               =cx for -1≤x≤2                =x+6 for x>2 To find the…