Q. 115 - Precalculus Enhanced with Graphing Utilities

Question

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

noting which graphs are the same.

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

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

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

Step-by-Step Solution

Answer

Similar Graphs are $y=\sqrt{x^2}$and $y=\left|x\right|$

Step 1. Given Information

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

Part (a) Step 1: Graph of y = x 2

Step 2. Graph of y = x

Step 3. Graph of y = x

Step 4. Graph of y = x 2

Part (b) Step 1. Same graph

The two functions with the same graph are $y=\sqrt{x^2}$ and $y=\left|x\right|$

They are same because $y=\sqrt{x^2}=\left|x\right|$

Part (c). Step 1. Explanation to graph y = x and y = x 2

These graphs are not same because range is not same.

Part (d). Step 1. Explanation to graph y = x 2 and y = x

Again, the graph is not same because the range is not same.