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I know this question may be stupid but I've been studying for my test tomorrow and I'm so frustrated, I can't figure this one out. if we have a square root function like this: $y = \sqrt{x}$ wouldn't $y$ be both $+2$ and $-2$ for $x = 4$, so shouldn't the graph look like a quadratic graph?

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By convention, $\sqrt x$ doesn't just mean any number whose square is $x$ -- it means the positive (or zero) number whose square is $x$.

In particular $\sqrt 4$ is $2$ and only that. It is true that $(-2)^2$ is also $4$, but because $-2$ is not positive, it doesn't satisfy the condition for being $\sqrt 4$.

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Functions send each $x$ to only one $y$ value. So while it is true that the root of a real number can be positive or negative, in order to graph as a function, the convention is to choose the positive root.

However, you'll notice the inverse of the function $y= {x}^{1/2}$ should also be a parabola, as it is formed by reflecting over the line $y=x,$ but one turned on its side, so your intuition is not far off. Just remember the convention for your exam, and good luck.

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  • $\begingroup$ Thanks, I'm wondering why our teacher didn't say anything about this "convention", it's been driving me crazy for the past 2 hours. $\endgroup$ – Ashkan Dec 26 '14 at 20:28
  • $\begingroup$ The inverse of $f(x) = \sqrt{x}$ is $f^{-1}(x) = x^2, x \geq 0$. $\endgroup$ – N. F. Taussig Dec 31 '14 at 1:33
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In this case it is so called principal square root, i.e. the nonnegative root of equation $x^2=y$.

See also http://en.wikipedia.org/wiki/Square_root.

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