137. Construct the graph of the function $y=f(f(f(x)))$, if $f(x)=$ $=\frac{1}{1-x}$.
S o l u t i o n. The domain of the function $f(x)$ is all real numbers except 1. Further,

![](https://cdn.mathpix.com/cropped/2024_05_21_98932225530737b82071g-157.jpg?height=431&width=531&top_left_y=801&top_left_x=357)

Fig 31

$$
f(f(x))=\frac{1}{1-\frac{1}{1-x}}=\frac{x-1}{x}
$$

The domain of the obtained function consists of all real numbers except 1 and 0. Finally,

$$
f(f(f(x)))=\frac{1}{1-\frac{x-1}{x}}=x
$$

Thus, the given function coincides with the function $y=x$ in its entire domain.

Therefore, the graph of the given function is the graph of the function $y=x(x \neq 0, x \neq 1)$ (Fig. 31).