Where One Half Comes Fa19C2

1. **The problem is:** why do we choose $x=\frac{1}{2}$ in the equation $f(x)+f\left(1-\frac{1}{x}\right)=8x$? 2. **The key idea is to make the inputs repeat.** We want to plug in values of $x$ that create a cycle, so the same few function values appear again and again. 3. **Start with one value and see what happens.** If we choose $x=2$, then the inside becomes $$1-\frac{1}{2}=\frac{1}{2}.$$ So $2$ gives us the value $\frac{1}{2}$. 4. **Now ask: what value of $x$ would produce $2$ inside the same expression?** We want $$1-\frac{1}{x}=2.$$ Solve it: $$-\frac{1}{x}=1$$ $$\frac{1}{x}=-1$$ $$x=-1.$$ So $-1$ gives us $2$. 5. **Now ask: what value of $x$ would produce $-1$ inside the same expression?** We want $$1-\frac{1}{x}=-1.$$ Solve it: $$-\frac{1}{x}=-2$$ $$\frac{1}{x}=2$$ $$x=\frac{1}{2}.$$ So $\frac{1}{2}$ gives us $-1$. 6. **That is where $\frac{1}{2}$ comes from.** It is not a random choice. It is chosen because it completes the cycle: $$2 \to \frac{1}{2} \to -1 \to 2.$$ This makes the same three function values appear, which helps us solve the problem. 7. **In short:** We choose $x=\frac{1}{2}$ because we are looking for the value of $x$ that makes $$1-\frac{1}{x}=-1.$$ That is exactly how $\frac{1}{2}$ appears. **Final answer:** $\frac{1}{2}$ comes from solving $1-\frac{1}{x}=-1$.