A-Level数学Core3复合函数化简疑问：f(f(x))化简解析

Hey there, let's walk through this simplification step by step—once you see the trick with common denominators, it'll click right away!

First, you correctly found that $f(f(x))$ gives us the initial expression:
$$\frac{\frac{x}{1-x}}{1-\frac{x}{1-x}}$$

The key here is to eliminate the nested fractions by using a common denominator for the bottom part of the big fraction. Let's break it down:

1. Simplify the denominator first
   The denominator of the big fraction is $1 - \frac{x}{1-x}$. To subtract these two terms, rewrite the whole number $1$ as a fraction with the same denominator as the other term:
   $$1 = \frac{1-x}{1-x}$$
   Now subtract the fractions:
   $$1 - \frac{x}{1-x} = \frac{1-x}{1-x} - \frac{x}{1-x} = \frac{(1-x) - x}{1-x}$$
   That's where the $(1-x)-x$ in the textbook's step comes from!

2. Rewrite the big fraction as a multiplication problem
   Remember that dividing by a fraction is the same as multiplying by its reciprocal. So our original expression becomes:
   $$\frac{\frac{x}{1-x}}{\frac{(1-x)-x}{1-x}} = \frac{x}{1-x} \times \frac{1-x}{(1-x)-x}$$

3. Cancel out the common terms
   Notice that $(1-x)$ appears in both the numerator of the second fraction and the denominator of the first fraction. As long as $1-x \neq 0$ (i.e., $x \neq 1$, which is already excluded from the domain of the original function $f(x)$), we can cancel these terms out:
   $$\frac{x}{\cancel{1-x}} \times \frac{\cancel{1-x}}{(1-x)-x} = \frac{x}{(1-x)-x}$$

4. Simplify the final denominator
   Combine like terms in the denominator:
   $$(1-x)-x = 1 - x - x = 1 - 2x$$
   So we end up with the simplified form:
   $$\frac{x}{1-2x}$$

Just a quick side note: make sure to remember the domain restrictions here—$x$ can't be $1$ (since it breaks the original $f(x)$) or $\frac{1}{2}$ (since it breaks $f(f(x))$).

内容的提问来源于stack exchange，提问作者Tom