The question says:
Suppose that $f^{-1}$ is the inverse function of $f$ and $y=f^{-1}(-x)$ is the inverse function of $y=f(-x)$; prove $f(x)$ is an odd function.
Now I know one of the solution is to make $g(x)=-x$ and use the features of composite function. But what I can't understand is that when I let $t=-x$, I think $f^{-1}(t)$ is the inverse function of $y=f(t)$, but this seems to suggest that $f^{-1}$ is the inverse of $f$.
Having already substituted $t=-x$ in $y=f(-x)$, you cannot also substitute it in $y=f^{-1}(-x)$ as this would be a different substitution. In this second case, $x$ belongs to the co-domain rather than domain of $f$. In fact, let's swap the variables $x$ and $y$ immediately, so that $x$ is in the domain and $y$ is in the co-domain: $x=f^{-1}(-y)$, and so the correct substitution there is $t=-f^{-1}(-y)$ rather than "$y=f^{-1}(t)$".