Problem (Algebra):
A function $f$ has the property that $f\left(\frac{2 x+1}{x}\right)=x+6$ for all real values of $x \neq 0$. What is the value of $f(4) ?$

Solution:
We note that $\frac{2 x+1}{x}=\frac{2 x}{x}+\frac{1}{x}=2+\frac{1}{x}$.

Therefore, $\frac{2 x+1}{x}=4$ exactly when $2+\frac{1}{x}=4$ or $\frac{1}{x}=2$ and so $x=\frac{1}{2}$.

Alternatively, we could solve $\frac{2 x+1}{x}=4$ directly to obtain $2 x+1=4 x$, which gives $2 x=1$ and so $x=\frac{1}{2}$.

Thus, to determine the value of $f(4)$, we substitute $x=\frac{1}{2}$ into the given equation $f\left(\frac{2 x+1}{x}\right)=x+6$ and obtain $f(4)=\frac{1}{2}+6=\frac{13}{2}$.

Final answer: $\frac{13}{2}$