[中學] 96基隆教甄 Q.21
※ 引述《JohnMash (John)》之銘言：
※ 引述《ruikim (倒數10)》之銘言：
: 1、在極坐標中，方程式r=cscΘ(cscΘ+cotΘ)的圖形為何？
: Ans:一個拋物線
: 2、設f為從{1,2,3,4} 映至{1,2,3,4} 的函數,滿足:f(f(x))=f(x),則這樣的函數有幾個?
: Ans:41
: 困擾了很久，麻煩大家了!!
if f(a)=a then f(f(a))=f(a)
if f(a)=b≠a then f(f(a))=f(b)=f(a)=b
That is, there exists at least one number n such that f(n)=n
(a) exact one number such that f(x)=x
for example, f(1)=1, f(2)=f(3)=f(4)=1
so there are 4 functions.
(b) exact two numbers such that f(x)=x
for example, f(1)=1, f(2)=2,
then [f(3)=f(4)=1] or [f(3)=1, f(4)=2] or [f(3)=2, f(4)=1]
or [f(3)=f(4)=2]
so there are C(4,2)*4=24 functions
(c) exact three numbers such that f(x)=x
for example, f(1)=1, f(2)=2, f(3)=3
then [f(4)=1] or [f(4)=2] or [f(4)=3]
so there are C(4,3)*3=12 functions
(d) exact four numbers such that f(x)=x
f(1)=1, f(2)=2, f(3)=3, f(4)=4
so there is 1 function
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