1. Let A, B be two non-empty sets and f: A + B a function. We say that f satisfies the o-property if VC+0.Vg, h: C + A, fog=foh=g=h. Prove that f is injective if and only if f satisfies the o-property.
13.1.11. Problem. Let f(x) = x and g(x) = 0 for all x ∈ [0,1]. Find a function h in B([0,1]) such that du(f,h) = du(f,g) = du(g,h). (3 problems) 13.2.6. Problem. Given in each of the following is the nth term of a sequence of real valued functions defined on (0, 1]. Which of these converge pointwise on (0, 1]? For which is the convergence uniform? (a) a z" (b) z+ nr. (c) a+ re-na 13.2.7. Problem. Given in...
Let f be a function defined as follows: 1 ?:Q−{0}→R, ?(?)=1− . ? Determine the set ?(?) ?h??? ????h????????? Q ??????? ?={?: ?=?, 1 Write down the set ?(?) by listing the elements as well as in the descriptive form ?∈Z−{0}}
Suppose f is a continuous and differentiable function on [0,1] and f(0)= f(1). Let a E (0, 1). Suppose Vr,y(0,1) IF f'(x) 0 and f'(y) ±0 THEN f'(x) af'(y) Show that there is exactly f(ax) and f'(x) 0 such that f(x) one Hint: Suppose f(x) is a continuous function on [0, 1] and f(0) x € (0, 1) such that f(x) = f(ax) f(1). Let a e (0,1), there exists an Suppose f is a continuous and differentiable function on...
For f(x) = 2-x and g(x) = 2x2 +x+6, find the following functions. a. (fog)x); b. (gof)(x); c. (fog)(2); d. (gof)(2) a. (fog)(x) = 0 (Simplify your answer.) b. (gof)(x) = 0 (Simplify your answer.) c. (fog)(2)- d. (gof)(2)-
1 Given f(x) = 5x² - 4 and g(x) = = 6 - find the following expressions. (a) (fog)(4) (b) (gof)(2) (c) (f o f)(1) (d) (gog)(0) (a) (fog)(4) = (Simplify your answer.) (b) (gof)(2) = (Simplify your answer.) (c) (f o f)(1) = (Simplify your answer.) (d) (g og)(0) = (Simplify your answer.)
Q Evaluate each expression using the graphs of y=f(x) and y=g(x) shown below. (a) (gof)(-1) (b) (gof)(0) (c) (fog) -1) (d) (fog)(4) (a) (gof)(-1)=0 (Simplify your answer.) (b) (gof)(0) = (Simplify your answer.) (C) (fog)(-1)=0 (Simplify your answer.) (d) (fog)(4)=0 (Simplify your answer.) y() 6 od 2- 4 y (K)
- Let V be the vector space of continuous functions defined f : [0,1] → R and a : [0, 1] →R a positive continuous function. Let < f, g >a= Soa(x)f(x)g(x)dx. a) Prove that <, >a defines an inner product in V. b) For f,gE V let < f,g >= So f(x)g(x)dx. Prove that {xn} is a Cauchy sequence in the metric defined by <, >a if and only if it a Cauchy sequence in the metric defined by...
Q 5. Write the following partial function f: Z4 → Z4 in table form. f = {(0,1), (1, 1), (2, 1), (3, 1)} Is f a total function? Explain why or why not.
4. For each n EN let fn: [0,1]R be given by if xE(0, otherwise fn(x) = (a) Find the function f : [0, 1] R to which {fn} converges pointwise. fn. Does {6 fn} converge to (b) For each n EN compute (c) Can the convergence of {fn} to f be uniform? 4. For each n EN let fn: [0,1]R be given by if xE(0, otherwise fn(x) = (a) Find the function f : [0, 1] R to which {fn}...