I am having trouble in graphing a particular type of functions where a function is divided piecewise, and for some pieces we have to be draw maximum part and for some we have to draw minimum part, for example,
Graph $g(x)$: $$ \text { Let } f(x)=x^{2}-2|x| \text { and } g(x)=\left\{\begin{array}{ccc} \min \{f(t)\}, & -2 \leq t< 0, & -2 \leq x<0 \\ \max \{f(t)\}, & 0 \leq t \leq 2, & 0 \leq x \leq 2 \\ f(x) & x>2 \end{array}\right. $$
Can anyone explain the concepts behind graphing such type of functions?
PS: Please provide a source for theory of graphing such type of functions.
The first step is find $\min\{f(t)\}$ on $-2 \leq t < 0$ and $\max\{f(t)\}$ on $0 \leq t \leq 2$. Let's call these quantities $m$ and $M$ respectively. Among other methods, let's graph $f$ to find $m$ and $M$ (since the question is interested in graphing). Below is a graph of $f$ (I used Desmos for the graphing):
We quickly see $m = -1$ and $M = 0$. Using this information, the definition of $g(x)$ becomes: $$ g(x) = \begin{cases} -1 & -2 \leq x < 0, \\ 0 & 0 \leq x \leq 2, \\ f(x) & x > 2. \end{cases} $$ This definition is easier to visualize (we have eliminated the parameter $t$), and graphing this, we have:
(sorry if the blue lines are a bit hard to see)