The problem asks us to identify which of the given functions is the same as the function $y = x$, where $x \in \mathbb{R}$. The options are: A. $y = |x|$ B. $y = \sqrt{x^2}$ C. $y = \sqrt[3]{x^3}$ D. $y = (\sqrt{x})^2$

1. Problem Description

The problem asks us to identify which of the given functions is the same as the function $y = x$, where $x \in \mathbb{R}$. The options are:

A. $y = |x|$

B. $y = \sqrt{x^2}$

C. $y = \sqrt[3]{x^3}$

D. $y = (\sqrt{x})^2$

2. Solution Steps

We need to analyze each option to determine which one is equivalent to $y = x$ for all real numbers $x$.

A. $y = |x|$. The absolute value function is defined as:

$|x| = x$ if $x \ge 0$

$|x| = -x$ if $x < 0$

Therefore, $y = |x|$ is not the same as $y = x$ for all $x \in \mathbb{R}$. For example, if $x = -1$, then $y = |-1| = 1 \ne -1$.

B. $y = \sqrt{x^2}$. Since the square root function returns the non-negative square root, we have:

$\sqrt{x^2} = |x|$

This is because if $x \ge 0$, then $\sqrt{x^2} = x$, and if $x < 0$, then $\sqrt{x^2} = -x$.

Thus, $y = \sqrt{x^2} = |x|$, which is not the same as $y = x$ for all $x \in \mathbb{R}$.

C. $y = \sqrt[3]{x^3}$. The cube root of a number $x$ is the value that, when cubed, gives $x$. This is true for all real numbers.

$\sqrt[3]{x^3} = x$ for all $x \in \mathbb{R}$.

Therefore, $y = \sqrt[3]{x^3}$ is the same as $y = x$ for all $x \in \mathbb{R}$.

D. $y = (\sqrt{x})^2$. First, we need to consider the domain of this function. The square root function $\sqrt{x}$ is only defined for non-negative values of $x$, i.e., $x \ge 0$. If $x \ge 0$, then $(\sqrt{x})^2 = x$. However, this function is not defined for $x < 0$. Therefore, this is not the same function as $y=x$ where $x \in \mathbb{R}$.

3. Final Answer

C