Question

77

想請問2,3(a),4,5(b)的詳解
謝謝🙏

2. (12%) Find
∞
[-1,1] (a) Ui=1
-120 №b) n (1-1)
(6) = 1
11.
[1]) = [1-1,1]
(c)
i=1
$(dina (i,∞)
i=1

3. (18%) Suppose A = {x, y} and B = {x, {x}}. Mark each of the following statements TRUE or
FALSE. (每小題僅需圈選T或 F 即可)
(a) [OF]
-6 L(b) (DF]
LC) [T
d) TE
Lle) [DF]
4) [TA
x≤B.
Ø E P(B).
≤A-B. X ≤A-B
|P(A)1 = 4:1 +₁5x3 593.5x²4]} | = 4
{{*}} = P(B). {{*},{*}. [***]]] =
{({x},{x})}≤ AXA.

2
$1. [1]
2 4 and C
-8
[06] Find a matrix B such that AB = Ct or prove that
AB= ct
t 4
B = C² A¹ = [ ²1 ] ² [ + ]
3
5. (10%) Suppose f: R → R and g: R → R where g(x) = 2x + 1 and gof(x) = 2x + 11.
(a) Find f.
-5
ial gofix =2xtil
Fix)=x+5
(b)
(b) Find g-¹.
(8%) Suppose A
no such matrix exists.
2 (F(x)) + ( + 2X+1
2(X)) = 2x+10
of thoro recurrence relations and initial conditions.

離散

Answer 1

線上上課都在線

以下僅為說明
2-1由於i=1之後的集合均為[-1,1]的真子集，所以答為[-1,1]
2-2因為1-1/i<1-1/(i+1)對正數i恆成立, 又不含端點(即1)，所以無限交集為空集合
2-3同上題，但包含端點，所以交集為{1}
2-4=(1,∞)∩(2,∞)∩(3,∞)∩...={}
pf:設k為2-4所求集合裡的一個元素（k∈R, k>1)
但因為若如此，k也必須在([k]+1,∞)裡面，而上述顯然不成立
所以k不能在所求集合裡，又根據假設，所求為{}

3-a因為x不為集合，所以該表達式不成立

4 B=A^(-1)C^t=(1/2){4,-5,-2,3}{2,0,1,6}={3/2,-15,(-1/2),9}
{}內矩陣{a,b,c,d}

5要求反函數，把g(x)改成g^(-1) (x)，並與x對調，需注意定義域與(值域or對應域)也要對調
簡單計算x=2g^(-1) (x)+1
g^(-1) (x)=(x-1)/2
g^(-1): R->R where g^(-1) (x)=(x-1)/2