\(f\left(x\right)+3f\left(\frac{1}{x}\right)=x^2\)
+ Thay \(x=2\) vào hàm số \(f\left(x\right)\) ta được:
\(f\left(2\right)+3f\left(\frac{1}{2}\right)=2^2\)
\(\Rightarrow f\left(2\right)+3f\left(\frac{1}{2}\right)=4\) (1).
+ Thay \(x=\frac{1}{2}\) vào hàm số \(f\left(x\right)\) ta được:
\(f\left(\frac{1}{2}\right)+3f\left(\frac{1}{\frac{1}{2}}\right)=\left(\frac{1}{2}\right)^2\)
\(\Rightarrow f\left(\frac{1}{2}\right)+3f\left(2\right)=\frac{1}{4}\)
\(\Rightarrow3f\left(2\right)+f\left(\frac{1}{2}\right)=\frac{1}{4}\)
\(\Rightarrow3.3f\left(2\right)+3f\left(\frac{1}{2}\right)=\frac{1}{4}.3\)
\(\Rightarrow9f\left(2\right)+3f\left(\frac{1}{2}\right)=\frac{3}{4}\) (2).
Lấy (2) trừ đi (1) ta được:
\(9f\left(2\right)-f\left(2\right)+3f\left(\frac{1}{2}\right)-3f\left(\frac{1}{2}\right)=\frac{3}{4}-4\)
\(\Rightarrow8f\left(2\right)+0=-\frac{13}{4}\)
\(\Rightarrow8f\left(2\right)=-\frac{13}{4}\)
\(\Rightarrow f\left(2\right)=\left(-\frac{13}{4}\right):8\)
\(\Rightarrow f\left(2\right)=-\frac{13}{32}.\)
Vậy \(f\left(2\right)=-\frac{13}{32}.\)
Chúc bạn học tốt!