Proving two statements are the same (with words).

[angela107]

Homework Statement

Are $(\sqrt{x})^2$ and $\sqrt{x^2}$ always equal for all real numbers x?

Relevant Equations

n/a

As the question asks, I believe this statement is true. At least, technically. It is important to consider the domain of the function. Yes, $sqrt(x)^2 = x$, but $sqrt(x)$ is only defined for nonnegative $x$, whereas $sqrt(x^2)$ is defined for all $x$, since $x^2$ is always nonnegative. The two functions are actually equal where they are both defined.

My question is, have my justified my solution well enough for a reader to understand what I'm saying?

 

[Mark44]

Homework Statement:: Are $(\sqrt{x})^2$ and $\sqrt{x^2}$ always equal for all real numbers x?
Relevant Equations:: n/a

As the question asks, I believe this statement is true. At least, technically. It is important to consider the domain of the function. Yes, $sqrt(x)^2 = x$, but $sqrt(x)$ is only defined for nonnegative $x$, whereas $sqrt(x^2)$ is defined for all $x$, since $x^2$ is always nonnegative. The two functions are actually equal where they are both defined.

My question is, have my justified my solution well enough for a reader to understand what I'm saying?

First off, those are expressions, not statements.
The question asks whether $(\sqrt{x})^2$ and $\sqrt{x^2}$ are always equal for all real numbers x.
Are the two expressions equal for, say, x = 1?

Keep in mind that where both expressions are defined is not necessarily all real numbers.