函數的求法
若函數\(f\)滿足：\(\cases{\displaystyle f(xy)=f(x)f(\frac{3}{y})+f(y)f(\frac{3}{x})\cr f(1)=\frac{1}{2}}\)
求滿足上述條件的所有函數\(f(x)=\)[u] [/u]。
請問此題如何作答,謝謝!
回復 1# lungxo 的帖子
\(\begin{align}& f\left( 1 \right)=\frac{1}{2} \\
& f\left( xy \right)=f\left( x \right)f\left( \frac{3}{y} \right)+f\left( y \right)f\left( \frac{3}{x} \right) \\
\end{align}\)
\(x=y=1\)代入，得\(f\left( 3 \right)=\frac{1}{2}\)
\(y=1\)代入，得\(f\left( x \right)=f\left( \frac{3}{x} \right)\)
\(y=\frac{3}{x}\)代入，得\({{f}^{2}}\left( x \right)+{{f}^{2}}\left( \frac{3}{x} \right)=\frac{1}{2}\)
故\(f\left( x \right)=\frac{1}{2}\)
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