Răspuns :
a)
[tex]\it x*y = \dfrac{1}{10}xy -(x+y) +20 = \dfrac{1}{10}xy \ -\ ^{10)}(x+y) +\ ^{10)}10 + 10 = \\ \\ \\ = \dfrac{xy - 10x-10y+100}{10} +10 = \dfrac{y(x-10) -10(x-10)}{10} +10 = \\ \\ \\ = \dfrac{(x-10)(y-10)}{10} +10 = \dfrac{1}{10}(x-10)(y-10) +10 [/tex]
b)
[tex]\it x*y \ \textless \ \dfrac{101}{10} \Rightarrow x*y \ \textless \ 10 + \dfrac{1}{10} \Rightarrow \\ \\ \\ \Rightarrow \dfrac{1}{10}(x-10)(y-10) +10\ \textless \ 10+\dfrac{1}{10} |_{-10} \Rightarrow \\ \\ \\ \Rightarrow \dfrac{1}{10}(x-10)(y-10) \ \textless \ \dfrac{1}{10} |_{\cdot 10} \Rightarrow (x-10)(y-10) \ \textless \ 1[/tex]
Avem 3 cazuri:
[tex]\it I)\ y=10 \Rightarrow x \in \mathbb{R} \\ \\ \\ II)\ y\ \textless \ 10 \Rightarrow x-10 \ \textgreater \ \dfrac{1}{y-10} |_{+10} \Rightarrow x \ \textgreater \ \dfrac{10y-99}{y-10} \\ \\ \\ II)\ y\ \textgreater \ 10 \Rightarrow x-10 \ \textless \ \dfrac{1}{y-10} |_{+10} \Rightarrow x \ \textless \ \dfrac{10y-99}{y-10}[/tex]
c)
Un factor al produsului din enunț este:
[tex]\it log_2{1024} = log_2{2^{10}} = 10log_2 2 = 10 \\ \\ \\ x*10 = \dfrac{1}{10} (x-10)(10-10) +10 = 0+10=10 \\ \\ \\ 10*y = \dfrac{1}{10}(10-10)(y-10)+10 = 0+10= 10 [/tex]
Deci, valoarea produsului din enunț este egală cu 10.
