Example 1 The function $f(x)$ defined on $\mathrm{R}^{+}$ satisfies the relation $f(x)=f\left(\frac{1}{x}\right) \lg x + 1$, find $f(x)$.
Solution: By substituting $\frac{1}{x}$ for $x$ in the given expression, we get $f\left(\frac{1}{x}\right)=f(x) \lg \frac{1}{x}+1$. From $\left\{\begin{array}{l}f(x)=f\left(\frac{1}{x}\right) \lg x+1 \\ f\left(\frac{1}{x}\right)=f(x) \lg \frac{1}{x}+1\end{array}\right.$, eliminating $f\left(\frac{1}{x}\right)$ yields $f(x)=\frac{1+\lg x}{1+\lg ^{2} x}$. Upon verification, the above $f(x)$ is a solution to the original equation.