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

Question

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

plomet6a · Accepted Answer

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 points out, the formula

(x^{a})^{b} = x^{a b}

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.

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

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