f(x) = min{g1(x), g2(x).......... } or max{g1(x), g2(x).......... } - Practice Questions & MCQ

$f(x)=\max \left\{g_1(x), g_2(x)\right\}$

$\because \mathrm{f}(\mathrm{x})= \begin{cases}g_1(x), & g_1(x)>g_2(x) \\ g_2(x), & g_2(x)>g_1(x)\end{cases}$

First draw the graph of both functions, $g_1(x)$ and $g_2(x)$ and find their points of intersection.
Then find any two consecutive points of intersection.
In between these points if $\mathrm{g}_1(\mathrm{x})>\mathrm{g}_2(\mathrm{x})$, then in order to draw the graph of $\max \left\{\mathrm{g}_1(\mathrm{x}), \mathrm{g}_2(\mathrm{x})\right\}$ take the graph of $y=g_1(x)$ and delete the graph of $y=g_2(x)$. But if in between these points if $g_1(x)$ $<\mathrm{g}_2(\mathrm{x})$, then in order to draw the graph of $\max \left\{\mathrm{g}_1(\mathrm{x}), \mathrm{g}_2(\mathrm{x})\right\}$ take the graph of $\mathrm{y}=\mathrm{g}_2(\mathrm{x})$ and delete the graph of $\mathrm{y}=\mathrm{g}_1(\mathrm{x})$

$f(x)=\min \left\{g_1(x), g_2(x)\right\}$

$\because \mathrm{f}(\mathrm{x})= \begin{cases}g_1(x), & g_1(x)<g_2(x) \\ g_2(x), & g_2(x)<g_1(x)\end{cases}$

In this case, take the minimum of $\mathrm{g} 1(\mathrm{x})$ or $\mathrm{g} 2(\mathrm{x})$ between any two consecutive points of intersection.
For example

$
f(x)=\max \left\{x, x^2\right\}
$
Draw the graph of $y=x$ and $y=x^2$

From the graph

For $x \in(-\infty, 0)$, the graph of $y=x^2$ lies above the graph of $y=x$, i.e. $x^2>x$, so we keep $y=x^2$ graph and delete $y=x$ graph in this interval

For $x \in(0,1)$, the graph of $y=x$ lies above the graph of $y=x 2$, i.e. $x>x^2$ so we keep $y=x$ graph and delete $y=x^2$ graph in this interval

For $x \in(1, \infty)$, the graph of $y=x^2$ lies above the graph of $y=x$, i.e. $x^2>x$, so we keep $y=x^2$ graph and delete $y=x$ graph in this interval

Hence, we have

$
\mathrm{f}(\mathrm{x})=\left\{\begin{array}{cc}
x^2, & x<0 \\
x, & 0 \leq x \leq 1 \\
x^2, & x>0
\end{array}\right.
$
for $f(x)=\min \left\{x, x^2\right\}$, we have
$
\mathrm{f}(\mathrm{x})=\left\{\begin{array}{cc}
x, & x<0 \\
x^2, & 0 \leq x \leq 1 \\
x, & x>0
\end{array}\right.
$