Why is the plot of $(x^2)^{1/2}$ not the same as the graph of $x$?

Question

Simple question: Why is the plot of $(x^2)^{1/2}$ different from the plot of $x$? I understand that the answer has to do with how we define absolute values and perhaps the order of operations, but doesn't $(x^a)^b = x^{ab}$? I feel like I'm missing some nuance about exponent algebra. Any clarification is helpful.

Answer 1

The square root function is, as a function, single-valued. And it’s defined to be always nonnegative.

In point of fact, one way of defining the absolute value function is: $|x|=\sqrt{x^2}$.

As to your specific question, as @lonzaleggiera points out, the formula $(x^a)^b=x^{ab}$ is valid only for nonnegative bases $x$. There’s another set of situations where the formula is valid, namely when both $a$ and $b$ are integers.

Let me get up on my high horse and complain that one of the many faults of high-school mathematics teaching is that insufficient attention is paid to the importance of the domain of functions, and similarly, to the domain of validity of rules such as the one you are asking about.

Answer 2

Whenever you have square roots, in this case $\sqrt{x^2}$, you have to be careful about negatives. Namely, here you know that your final answer is nonnegative, so you definitely can't have $f(x) = x$ for $x < 0$ for the function you named.

Answer 3

A simple example with a concrete number can be of help:$$\sqrt {(-2)^2}=\sqrt 4=2$$. The big thing here is that the squaring loses the sign information of the input, both positive and negatives square to the same output. Then the square root function only outputs the positive square root. Thus, the combination strips the sign off the input, giving us $$\sqrt {x^2}=|x|$$