Answer
See below.
Work Step by Step
(a)
Now, to graph the functions \[y=\sqrt{{{x}^{2}}},\,y=x,\,y=\left| x \right|\] and \[y={{\left( \sqrt{x} \right)}^{2}}\] we use a graphing calculator.
(b)
From part (a),
When $x<0$, \[y=\sqrt{{{x}^{2}}}=-x\] \[y=\left| x \right|=-x\]
Thus, the points on the graphs \[y=\sqrt{{{x}^{2}}}\]and \[y=\left| x \right|\] are the same.
When $x>0$, \[y=\sqrt{{{x}^{2}}}=x\] \[y=\left| x \right|=x\]
Thus, the points on the graphs \[y=\sqrt{{{x}^{2}}}\]and \[y=\left| x \right|\] are the same.
When $x=0$, \[y=\sqrt{{{x}^{2}}}=0\] \[y=\left| x \right|=0\]
Thus, the point $\left( 0,0 \right)$ is the same on the graphs \[y=\sqrt{{{x}^{2}}}\] \[y=\left| x \right|\]
Therefore, for any $x$, the graphs \[y=\sqrt{{{x}^{2}}}\]and \[y=\left| x \right|\]are the same.
(c)
From part (a),
The domain of \[y=x\] is all real numbers $x$, and the domain of \[y={{\left( \sqrt{x} \right)}^{2}}\] is $x\ge 0$.
Thus, the domains of the functions \[y=x\] and \[y={{\left( \sqrt{x} \right)}^{2}}\] are different.
Therefore, the graphs \[y=x\] and \[y={{\left( \sqrt{x} \right)}^{2}}\] are not the same.
(d)
From part (a),
When $x<0$, \[y=\sqrt{{{x}^{2}}}=-x\].
But, for $x<0\,\,\Rightarrow \,y=x$.
Thus, the points on the graphs of \[y=\sqrt{{{x}^{2}}}\] and \[y=x\]are different for $x<0$.
Therefore, for $x<0$, the points on graphs \[y=\sqrt{{{x}^{2}}}\] and \[y=x\] are not the same.
Hence, the graphs of \[y=\sqrt{{{x}^{2}}}\] and \[y=x\] are not the same.
