I'm preparing for a local math competition/olympiad, so I've been researching for past exams, and I've found this problem:
Consider the lineal functions $f$ and $g$ such that $f(2)=1$,$\;g(2x)=-2f(x)\,$ and $\;{g}^{-1}(x)=g(x)$. The value of $f(2013)$ is given by:
a) 2
b) 2012
c) 2013
d) 2014
My first intuition was to consider $x=2$, where $$g(4)=-2f(2)=-2$$ So if I were to graph $g(x)$ (which is linear), the line would go through the point $(4,-2)$.
Now, I know I need to use the fact that $\;{g}^{-1}(x)=g(x)$, but I'm not exactly sure how to interpret it. I'm pretty sure it means that $g(x)$ and its inverse have the same output. But does this imply that $g(x)$ and $\;{g}^{-1}(x)$ are exactly the same function? Or can they be different functions which have the same output for any $x$? Is this last case even possible in linear functions?
I've read somewhere that this is called an involution. Either way, how could I use this fact to, say, determine another point in the graph of $g(x)$, and thus, determine $g(x)$? I know this is somewhat elementary, but I'm confused.
By the way, this is my first question here, so please bear with any formatting mistakes I've made.
If two functions have the same output and the same domain, they are the same function.
You know that $g(x)=kx+n$, and that $g^{-1}(x) = g(x)=kx+n$.
Now, using the properties of inverse functions, you also know that for each $x$, you have $x = \mathrm{id}(x) = (g\circ g^{-1})(x) = (g\circ g)(x)=g(g(x))$, and this equation should give you a lot of information.