The problem asks us to find an expression for $y$ such that the given condition "jGty = x, where x is a real number" holds true. The options are: A. $Ay = z$ B. $y = \sqrt{x^2}$ C. $y = \sqrt[3]{x^3}$ D. $y = (\sqrt{x})^2$

1. Problem Description

The problem asks us to find an expression for $y$ such that the given condition "jGty = x, where x is a real number" holds true. The options are:

A. $Ay = z$

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

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

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

2. Solution Steps

Let's analyze each option:

A. $Ay = z$: This expression relates $A$, $y$, and $z$, but it doesn't give us $y$ as a function of $x$. So this is not the correct answer.

B. $y = \sqrt{x^2}$: Since $x$ is a real number, $\sqrt{x^2} = |x|$. Therefore, $y = |x|$. This is not the same as $y = x$ in general because $|x|$ is always non-negative, whereas $x$ can be negative. So this is not the correct answer.

C. $y = \sqrt[3]{x^3}$: Since $x$ is a real number, $\sqrt[3]{x^3} = x$. Therefore, $y = x$. This is a possible answer.

D. $y = (\sqrt{x})^2$: For this expression to be defined, $x$ must be non-negative. If $x \geq 0$, then $(\sqrt{x})^2 = x$. However, the problem states that $x$ is any real number, so $x$ can be negative. Therefore, this equation does not hold for all real numbers. So, this is not the correct answer.

Comparing option C to the problem, we can see that $y = x$ is the solution.

3. Final Answer

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