1.2.7 * Given that the graph of the function $f(x)=\frac{a x+b}{c x+d}$ coincides with the graph of its inverse function. What form should this function have? Here $a, b, c, d$ are constants, and $a, c$ are not both 0.
From $y=\frac{a x+b}{c x+d}$, we can derive $x=\frac{d y-b}{-c y+a}$. Therefore, the inverse function of $f(x)$ is $f^{-1}(x)=$ $\frac{d x-b}{-c x+a}$. According to the condition, $\frac{a x+b}{c x+d}=\frac{d x-b}{-c x+a}$, which means
$$
(c d+a c) x^{2}+\left(d^{2}-a^{2}\right) x-b(a+d)=0 .
$$

The left side of the above equation should be a zero polynomial, i.e., $c(a+d)=d^{2}-a^{2}=-b(a+d)=0$, which indicates that $a+$ $d=0$ or $a=d(\neq 0)$ and $b=c=0$. Therefore, $f(x)=\frac{a x+b}{c x-a}$ or $f(x)=x$.