Chapter 1.2, Problem 85AYU

Graph $y=\sqrt{x^2}$, $y=x$, $y=\left|x\right|$ and $y={\left(\sqrt{x}\right)}^2$, noting which graphs are the same.

Explain why the graphs of $y=\sqrt{x^2}$ and $y=\left|x\right|$ are the same.

Explain why the graphs of $y=x$ and $y={\left(\sqrt{x}\right)}^2$ are not the same.

Explain why the graphs of $y=\sqrt{x^2}$ and $y=x$ are not the same.

To determine

To graph: The functions $y=\sqrt{x^2},y=x,y=\left|x\right|$ and $y={\left(\sqrt{x}\right)}^2$, noting which graphs are the same.

Explanation of Solution

Given Information:

The functions $y=\sqrt{x^2},y=x,y=\left|x\right|$ and $y={\left(\sqrt{x}\right)}^2$

Graph:

Now, to graph the functions $y=\sqrt{x^2},y=x,y=\left|x\right|$ and $y={\left(\sqrt{x}\right)}^2$ by using graphing calculator.

For $y=\sqrt{x^2}$,

Use the steps below to graph the function:

Step 1: Press [Y=] key. Then, enter the function as $Y_1=\sqrt{X^2}$.

Step 2: Press [WINDOW] key and set the viewing window as below:

$\begin{array}{ll}X\min =-5\hfill & X\max =5X\text{scl}=1\hfill \\ Y\min =0\hfill & Y\max =5Y\text{scl}=1\hfill \end{array}$ $$.

Step 3: Then hit the [Graph] button to view the graph.

For $y=x$,

Use the steps below to graph the function:

Step 1: Press [Y=] key. Then, enter the function as $Y_1=X$.

Step 2: Press [WINDOW] key and set the viewing window as below:

$\begin{array}{ll}X\min =-5\hfill & X\max =5X\text{scl}=1\hfill \\ Y\min =0\hfill & Y\max =5Y\text{scl}=1\hfill \end{array}$

Step 3: Then hit the [Graph] button to view the graph.

For $y=\left|x\right|$,

Use the steps below to graph the function:

Step 1: Press [Y=] key. Then, enter the function as $Y_1=\left|X\right|$.

Step 2: Press [WINDOW] key and set the viewing window as below:

$\begin{array}{ll}X\min =-5\hfill & X\max =5X\text{scl}=1\hfill \\ Y\min =0\hfill & Y\max =5Y\text{scl}=1\hfill \end{array}$

Step 3: Then hit the [Graph] button to view the graph.

For $y={\left(\sqrt{x}\right)}^2$,

Use the steps below to graph the function:

Step 1: Press [Y=] key. Then, enter the function as $Y_1={\left(\sqrt{X}\right)}^2$.

Step 2: Press [WINDOW] key and set the viewing window as below:

$\begin{array}{ll}X\min =-5\hfill & X\max =5X\text{scl}=1\hfill \\ Y\min =0\hfill & Y\max =5Y\text{scl}=1\hfill \end{array}$

Step 3: Then hit the [Graph] button to view the graph.

Interpretation:

From the above graphs, the graphs of the functions $y=\left|x\right|$ and $y=\sqrt{x^2}$ are same.

To determine

Why the graphs of $y=\sqrt{x^2}$ and $y=\left|x\right|$ are same.

Answer to Problem 85AYU

Solution:

For any $x$, the graphs $y=\sqrt{x^2}$ and $y=\left|x\right|$ are same.

Explanation of Solution

Given Information:

Instruction to explain why the graphs of $y=\sqrt{x^2}$ and $y=\left|x\right|$ are same.

Explanation:

From part (a),

The graphs of $y=\sqrt{x^2}$ and $y=\left|x\right|$ are shown below:

From the above graphs, it is observed that

When $x<0$, then $y=\sqrt{x^2}=-x$, and $y=\left|x\right|=-x$.

Thus, the points on the graphs $y=\sqrt{x^2}$ and $y=\left|x\right|$ are same.

When $x>0$, then $y=\sqrt{x^2}=x$, and $y=\left|x\right|=x$.

Thus, the points on the graphs $y=\sqrt{x^2}$ and $y=\left|x\right|$ are same.

When $x=0$, then $y=\sqrt{x^2}=0$, and $y=\left|x\right|=0$.

Thus, the point $\left(0,0\right)$ is same on the graphs $y=\sqrt{x^2}$ and $y=\left|x\right|$.

Therefore, for any $x$, the graphs $y=\sqrt{x^2}$ and $y=\left|x\right|$ are same.

To determine

Why the graphs of $y=x$ and $y={\left(\sqrt{x}\right)}^2$ are not the same.

Answer to Problem 85AYU

Solution:

The domain of the graphs of $y=x$ and $y={\left(\sqrt{x}\right)}^2$ are different.

Explanation of Solution

Given Information:

The graphs of $y=x$ and $y={\left(\sqrt{x}\right)}^2$.

Explanation:

From part (a),

The graphs of $y=x$ and $y={\left(\sqrt{x}\right)}^2$ are shown below:

From the above graphs, it is observed that

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 same.

To determine

Why the graphs of $y=\sqrt{x^2}$ and $y=x$ are not the same.

Answer to Problem 85AYU

Solution:

For $x<0$, the points on graphs $y=\sqrt{x^2}$ and $y=x$ are not the same.

Explanation of Solution

Given Information:

The graphs of $y=\sqrt{x^2}$ and $y=x$.

Explanation:

From part (a),

The graphs of $y=\sqrt{x^2}$ and $y=x$ are shown below:

From the above graphs, it is observed that

When $x<0$, then $y=\sqrt{x^2}=-x$.

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.