MATH1050B/C Assignment7Due date:20-3-2017(1900hrs)
黄金大劫案电影Part1.Submit your work on at least Questions(3),(4),(5),(6),(7),(8).
1.Let A,B,C,D be ts,and f:A−→B,g:B−→C,h:C−→D be functions.
Prove that(h◦g)◦f=h◦(g◦f)as functions.
2.Let f,g:R−→R be functions defined by f(x)=
x2
1+x2
,g(x)=x−1for any x∈R.
(a)Compute the respective‘formulae of definition’of the functions g◦f,f◦g explicitly.
(b)Choo some x0∈R so that(g◦f)(x0)=(f◦g)(x0).
(c)Is it true that(g◦f)(x)=(f◦g)(x)for any x∈R?Justify your answer.
Remark.Hence we have dis-proved the statement below:
•Let A be a t,and f,g:A−→A be functions.g◦f=f◦g as functions.
霎时3.Let f:R−→R be the function defined by f(x)=x53−1for any x∈R.
(a)Verify that f is surjective(directly from the definition of surjectivity).
(b)Verify that f is injective(directly from the definition of injectivity).
4.Denote the interval(0,+∞)by I.Let f:I−→R be the function defined by f(x)=1
2 x−1x
for any x∈I.
(a)Verify that f is injective(directly from the definition of injectivity).
(b)Verify that f is surjective(directly from the definition of surjectivity).
Remark.It may help if you start by considering whether,for each b∈R,the equation b=f(u)with unknown u has any solution or not.
5.Let f:[0,9]−→R be the function defined by f(x)=−x+6
√x−5for any x∈[0,9].
(a)Show that f(x)=−(A−
√x)2+B for any x∈[0,9].Here A,B are some real constants who respective values you have to determine.
(b)Verify that f is injective,directly from the definition of injectivity.
(c)Is f surjective?Justify your answer directly from the definition of surjectivity.
6.Let f:(0,+∞)−→R be the function defined by f(x)=x2−1
x2+1
sin 1√x for any x∈(0,+∞).
(a)Verify that f is not injective.
(b)i.♦Verify that x2−1x2+1 ≤1for any x∈(0,+∞).
春运客流量鸡翅煲的做法Remark.A very simple answer can be obtained without using calculus.
ii.Apply the previous part,or otherwi,to verify that f is not surjective.
7.Let f:C−→C be the function defined by f(z)=¯z for any z∈C.
(a)Verify that f is injective.(b)Verify that f is surjective.
8.Let f:C\{0}−→C\{0}be the function defined by f(z)=z
¯z最美诗句经典语录
for any z∈C\{0}.
(a)Verify that f is not injective.(b)Verify that f is not surjective.
9.Let f:C\{0}−→C\{0}be the function defined by f(z)=z2热播的电视剧
¯z
for any z∈C\{0}.
(a)Verify that f(z)=
z3
|z|2for any z∈C\{0}.
(b)♦Is f injective?Justify your answer.
(c)♣Is f surjective?Justify your answer.
Part2.
1.♦We recall/introduce the definitions/notations:
•Let A,B,C,D be ts,and f:A−→C,g:B−→D be functions.Suppo f(x)=g(x)for any x∈A∩B.
Define the function f∪g:A∪B−→C∪D by
(f∪g)(x)= f(x)if x∈A
g(x)if x∈B
The function f∪g is called the union of the functions f,g.
Consider each of the statements below.Determine whether it is true or fal.Justify your answer with an appropriate argument.
勇敢怎么写(a)Let A,B,C,D be ts,and f:A−→C,g:B−→D be functions.Suppo f(x)=g(x)for any x∈A∩B.
Suppo f,g are surjective.Then f∪g:A∪B−→C∪D is surjective.
(b)Let A,B,C,D be ts,and f:A−→C,g:B−→D be functions.Suppo f(x)=g(x)for any x∈A∩B.
Suppo f,g are injective.Then f∪g:A∪B−→C∪D is injective.
2.♥Familiarity with the calculus of one variable is assumed in this question.
Let J be an open interval in R.Denote by C(J)the t of all real-valued continuous functions on J.Denote by C1(J) the t of all real-valued differentiables functions on J whofirst derivatives are continuous functions on J.
Define the function D:C1(J)−→C(J)by D(ϕ)=ϕ′for anyϕ∈C1(J).
For each a∈J,define the function I a:C(J)−→C1(J)by(I a(ψ))(x)= x aψ(t)dt for anyψ∈C(J)for any x∈J.
收到图片(a)i.Prove that((I a◦D)(ϕ))(x)=ϕ(x)−ϕ(a)for anyϕ∈C1(J)for any x∈J.
ii.Prove that((D◦I a)(ψ))(x)=ψ(x)for anyψ∈C(J)for any x∈J.
(b)i.Is the function I a◦D:C1(J)−→C1(J)surjective?Justify your answer.
ii.Is the function I a◦D:C1(J)−→C1(J)injective?Justify your answer.
3.We introduce/recall the definitions:
•Let n∈N.A degree-n polynomial with complex coefficients and with indeterminate z is an expression of the form a n z n+···+a1z+a0in which a0,a1,···,a n∈C and a n=0.
•A complex-valued function of one complex variable is called a degree-n polynomial function on C exactly when its‘formula of definition’is given by a degree-n polynomial with complex coefficients.
•Letζ∈C,and f(z)≡a n z n+···+a1z+a0be a polynomial with complex coefficients and with indeterminate z.ζis said to be a root of the polynomial f(z)in C if f(ζ)=0.
The statement(♯)below,first proved by Gauss,is known as the Fundamental Theorem of Algebra:
(♯)Every non-constant polynomial with complex coefficients(and with one indeterminate)has a root in C.
(a)♦Prove that the statement(♯)is logically equivalent to the statement(♭)below:
(♭)For any n∈N\{0},every degree-n polynomial function on C is surjective.
(b)♣Let n∈N\{0,1},a0,a1,···,a n∈C,with a n=0,and f:C−→C be the degree-n polynomial function defined
by f(z)=a n z n+···+a1z+a0for any z∈C.
Apply the Fundamental Theorem of Algebra,or otherwi,to prove that f is not injective.
Remark.Here you may also take for granted the Factor Theorem(who‘real version’you have already learnt at school and may be carried in verbatim to the‘complex situation’):
•Letα∈C,and p(z)be a degree-n polynomial(with complex coefficients).Suppoαis a root of p(z).
Then there is a degree-(n−1)polynomial q(z)(with complex coefficients)so that p(z)=(z−α)q(z)as
polynomials.
