求一道英文数学题!consider the linear function f(x)=ax+b.compute all ordered pairs (a,b) such that for all real numbers x,f(f(x))+f(x)=6x+60.
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求一道英文数学题!consider the linear function f(x)=ax+b.compute all ordered pairs (a,b) such that for all real numbers x,f(f(x))+f(x)=6x+60.
求一道英文数学题!
consider the linear function f(x)=ax+b.compute all ordered pairs (a,b) such that for all real numbers x,f(f(x))+f(x)=6x+60.
求一道英文数学题!consider the linear function f(x)=ax+b.compute all ordered pairs (a,b) such that for all real numbers x,f(f(x))+f(x)=6x+60.
呵呵,直接代入啊...f(f(x))=a(ax+b)+b=a^2x+ab+b.
f(f(x))+f(x)=a^2x+ax+ab+2b...要满足在任意情况下都等于右边6x+60.
则对应的系数要相等哦,对应项的.
a^2+a=6.a=2 或 a=-3
ab+2b=60 当a=2时,b=15..当a=-3时,b=-60..
所以满足条件的(a,b)为
(2,15) (-3,-60)
[a(ax+b)+b]+ax+b=6x+60
a^2x+ab+ax+2b=6x+60
(a^2+a)x+ab+2b=6x+60
=>a^2+a=6 and ab+2b=60
(a+3)(a-2)=6
a=-3 or a=2 and when a=-3 b=-60 when a=2 b=15
Plugging f(x)=ax+b in to the identity, we have
a(ax+b)+b+ax+b=6x+60.
or a*a+a=6 and ab+2b=60
thus a=2 b=15 or a=-3 b=-60
将等式整理得(a^2+a)x+ab+2b=6x+60
所以:a^2+a=6
ab+2b=60
解得:a=-3或a=2
b=-60或b=15
综上:(-3,-60),(2,15)
答案:(-3,-60)(2,15)
步骤:f(f(x))+f(x)=6x+60
a*(ax+b)+b+ax+b=6x+60
(a*a+a)x+(a*b+2b)=6x+60
a*a+a=6
a*b+2b=60
a、b不等于0
...
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答案:(-3,-60)(2,15)
步骤:f(f(x))+f(x)=6x+60
a*(ax+b)+b+ax+b=6x+60
(a*a+a)x+(a*b+2b)=6x+60
a*a+a=6
a*b+2b=60
a、b不等于0
由a*a+a=6
(a+3)(a-2)=0
a=-3或a=2
代入a*b+2b=60
a=-3 b=-60
a=2 b=15
收起
f(x)=ax+b
f(f(x))=a(ax+b)+b=a^2x+ab+b
f(f(x))+f(x)=a^2x+ab+b+ax+b=(a^2+a)x+ab+2b=6x+60
a^2+a=6,ab+2b=60
a=-3,b=-60
a=2,b=15
原式等于 a*(ax+b)+b+ax+b=6x+60
a*ax+ab+b+ax+b=6x+60
a*a+a=6 (a+1/2)*(a+1/2)=25/4 a=2或-3
ab+2b=60 b=15或-60
所以两个点 (2,15)(-3,-60)