Why doesn't the graph of sqrt(x) have output in the negative

>>8442215
No, the primary branch of the square root only takes positive values. You could in theory use the negative branch if you like but nobody does.

it is a convention, like it is a convention that functions are single valued
there's a difference between the solution set of an equation and the graph of a function.

>>8442215
A graph is made out of a function. A function associates one number with another number.

You're right, there are two numbers which squared give 4: -2 and 2. However, the square root of x is defined as the positive root for all numbers. So the graph of sqrt(x) is always above the x-axis as it is always positive by definition.

[math] \displaystyle
\sqrt {x^2} \ne \pm x, \quad \sqrt {x^2} = \left | x \right |
\\
\left | x \right | =
\left \{
\begin{align}
x, & \hspace{1em} x \geq 0 \\
-x, & \hspace{1em} x < 0
\end{align}
\right .
[/math]

>>8442299
this

>>8442215
sqrt(x) is a function, so it can only give one value, by convention the positive square root of x

>>8442277
What is your definition of graph?
We can graph many things which aren't functions.

>>8442362
Not OP but surely we can agree that the graph of T is the set {(x,T(x)):x in D(T)}?

>>8442362
Sorry, I didn't realize I was giving a class to MIT graduates.

>>8442277
I don't understand, in pre calc I studied all the conics (including the circumference) which aren't all functions. So why can't we draw the sqrt graph with two branches?

>>8442492
You *can* draw both but then it won't be a function. That is, you can graph the set of points (x, y) such that x^2 = y. But this is a relation, not a function. sqrt is a function.

>>8443861
*y^2 = x, that is.