if $f(x) = |3x - 1|$ so what is the sum of all $x$ such that $f(f(x)) = x$?

Question

Question: if $f(x) = |3x - 1|$ so the sum of all values of x that satisfies $f(f(x)) = x$ is?

alternatives:
a) $11/10$
b) $21/20$
c) $27/20$
d) $23/20$
e) $5/4$

My try: I've thought that I had to calculate $|3x - 1| = x$, since $f(f(x)) = f(|3x -1|)$. We'll have the answers $1/2$ and $1/4$, and the sum $3/4$. But doesn't match with the alternatives.

What I have to do??

Answer 1

Hint towards a quick and dirty solution:

Show that $ \pm (3 ( \pm (3x-1) -1 ) = \{9x-4 , -9x + 2, -9x+4, 9x-2\} $.
For each of these lines, solve for $\ell(x) = x$.
Then, verify if they are solutions to $f(f(x)) = x$, and take their sum

Note

• We might have extraneous solutions in the general case. For these values, we do not, and there are 4 solutions.
• (Esp in a competition setting, or when we only care about the values of the solution,) I find checking if they are indeed solutions an acceptable tradeoff against additionally determining the intervals which these lines correspond to. I find that the latter is much more time consuming, though it tends to be the approach that teachers prefer/use.