1. Let $y=f(x)$ be the exponential function $y=a^{x}$. Among the four points $P(1,1)$, $Q(1,2)$, $M(2,3)$, $N\left(\frac{1}{2}, \frac{1}{4}\right)$, the common points of the graph of the function $y=f(x)$ and its inverse function $y=f^{-1}(x)$ can only be point ( ).
(A) $P$
(B) $Q$
(C) $M$
(D) $N$
-1.D.
Since the inverse function of the exponential function $y=a^{x}(a>0, a \neq 1)$ is the logarithmic function $y=\log _{a} x$, and the logarithmic function passes through the point $(1,0)$, it cannot pass through points $P$ and $Q$.
If it passes through point $M$, then $3=a^{2}, a=\sqrt{3}$.
But $y=\log _{\sqrt{3}} 2 \neq 3$.
Taking $\frac{1}{4}=a^{\frac{1}{2}}$, we get $a=\frac{1}{16}$.
And $\log _{\frac{1}{16}} \frac{1}{2}=\frac{1}{4}$, so the common point can only be point $N$.