Răspuns :
[tex]x * y=\frac{100(x+y)}{x y+100}[/tex]
a)
Pentru a calcula 3*0 va trebui sa inlocuim pe x cu 3 si pe y cu 0
[tex]3 *0=\frac{100(3+0)}{3\times 0+100}=\frac{300}{100}=3[/tex]
b)
Calculam f(x*y)
[tex]f(x*y)=f(\frac{100(x+y)}{x y+100})=\frac{10-\frac{100(x+y)}{x y+100}}{10+\frac{100(x+y)}{x y+100}} =\frac{\frac{10xy+1000-100x-100y}{xy+100} }{\frac{10xy+1000+100x+100y}{xy+100}}[/tex]
Se reduc numitorii xy+100 si ne ramane:
[tex]f(x*y)=\frac{10xy+1000-100x-100y}{10xy+1000+100x+100y}[/tex]
Simplificam cu 10 fractia si ne ramane:
[tex]f(x*y)=\frac{xy+100-10x-10y}{xy+100+10x+10y}=\frac{x(y-10)-10(y-10)}{x(10+y)+10(y+10)} =\frac{(x-10)(y-10)}{(x+10)(y+10)}[/tex]
Acum vom calcula f(x)×f(y)
[tex]f(x)\times f(y)=\frac{10-x}{10+x} \times \frac{10-y}{10+y} =\frac{(x-10)(y-10)}{(x+10)(y+10)}=f(x*y)[/tex]
Observam ca f(x)×f(y)=f(x*y)
c)
Ne folosim de punctul b
f(x*y)=f(x)f(y)
Atunci f(x*x)=f(x)f(x)=(f(x))²
Din enunt x*x*x...*x=0, de 11 ori compus
Atunci f(x*x*x...*x)=f(0), de 11 ori compus
f(x*x*...*x)=(f(x))¹¹
(f(x))¹¹=f(0)
Calculam f(0)
[tex]f(0)=\frac{10-0}{10+0} =1[/tex]
Deci (f(x))¹¹=1 ⇒f(x)=1
[tex]\frac{10-x}{10+x} =1[/tex]
10-x=10+x
10-10=2x
x=0
Un exercitiu similar de bac gasesti aici: https://brainly.ro/tema/4711195
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