According to a problem that I am working on a fixed point is defined as a point $x$ that satisfies $$f(x) = x.$$
A problem is asking me to find the fixed point of the inverse of $f(x)$, and the explanation tells me to find the inverse function first, and later use the definition.
Is this step necessary? I want to say no because the inverse of a function has an image that is reflected about the line $y=x$, so finding the fixed point of the original function should give you the fixed point of the inverse as well.
Is there an exception that I am not aware about?
Best Answer
I believe the OP meant to say "if $f: R \rightarrow R$ is a function, then the graph of its inverse is simply the graph of $f$ reflected about the line $y = x$."
I'm going to assume that the original function is bijective (i.e., "one-to-one and onto"), so that it actually has an inverse function, rather than an inverse relation.
Let $g$ be the inverse of the function $f$. Suppose that $x_0$ is a fixed point of $g$. That means that
$g(x_0) = x_0$
Applying the function $f$ to both sides of this equation, you get
$f(g(x_0)) = f(x_0)$
Since $f$ and $g$ are inverses, we know that $f(g(x_0)) = x_0$, so this equation can be rewritten
$x_0 = f(x_0)$.
In other words, I've shown that any fixed point of $g$ is also a fixed point of $f$. (Essentially the same proof shows that fixed points of $f$ are fixed points of $g$ as well.) So if the problem asks you to find fixed points of the inverse of $f$, you might as well just look for fixed points of $f$, i.e., don't bother with trying to invert it.
-John