Let $$f(x) = \begin{cases} x+2, -3\leq x \lt-1 \\ x-1,-1\leq x \lt3 \end{cases}$$
I had to find $f(f(x)$
I defined $f(f(x)$ to be:
$$f(f(x)) = \begin{cases} f(x)+2, -3\leq f(x) \lt-1 \\ f(x)-1,-1\leq f(x) \lt3 \end{cases}$$
Finally it evaluated to be $$f(f(x)) = \begin{cases} x+1, -3\leq f(x) \lt0 \\ x-2,0\leq f(x) \lt3 \end{cases}$$
But by graphing on a graphing application Desmos, I got the following graph:
(purple lines indicate $f(f(x))$)
Link: https://www.desmos.com/calculator/knrx71as8b
Isn't the graph wrong? How can $f(f(x))$ return $x+2$ ?
You typed it in wrong: you defined $f(x)$ as $x+1$ on the interval $-3\le x\le -1$ in Desmos, where you meant $x+2$. When you fix this typo, the graph is correct.
In your post above, your final evaluation should read $$f(f(x)) = \begin{cases} x+1, &\text{if } {-}3\leq x \lt0 \\ x-2, &\text{if }0\leq x \lt3 \end{cases}$$ (you had $f(x)$ in place of $x$ on the right-hand side).