I can justify why the answer might be no, but also why it might be yes, so which line of reasoning is correct?
I. $x=\sqrt{y}$ does not represent y as a function of x because if $x$ is a negative number, then no corresponding y-value can be made, and by definition an equation is a function only if each x-value has exactly one corresponding y-value.
or
II. $x = \sqrt{y}$ does represent y as a function of x because solving the equation for $y$ yields $y=x^2$ with the domain restricted to [0,$\infty$), and graphing this yields half a parabola that passes the Vertical Line Test.
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$\begingroup$Your option 2 is correct, as long as we add a bit of clarification:
- The equation $x = \sqrt{y}$ represents $y$ as a function of $x$ on the domain $x \ge 0$.
Functions only make sense when they are defined on a particular domain, i.e. a set of input values. Once we make that clear, then we can check that your given equation does indeed meet the requirements of a function, most importantly that for any given $x$ value, there is only one valid $y$ value.
$\endgroup$ 3 $\begingroup$Does the expression $x=\sqrt{y}$ represent a function? The answer is yes. A function is a mapping of values from one set of real numbers (typically, real numbers), $A$, to another set of real numbers, $B$. The important thing with functions is that each element in $A$ should be associated with exactly one element in $B$. Is this true of $x=\sqrt{y}$? Well, $y$ can assume all possible nonnegative numbers and each such number when put throught the square root function will spit out exactly one $x$ value associated with that particular $y$ value that you plug in. Does this meet the criteria that make a relation of two sets of numbers a function? Yes, it does. So, $x=\sqrt{y}$ as a function of $y$ is a function with the domain $x\ge 0$ since the process of square-rooting can only give you numbers that are nonnegative and the range $y\ge 0$ because the only values you are allowed to take the square root of are those that are nonnegative. Again, I assume we're only working with real numbers.
$\endgroup$ $\begingroup$There are two different uses of the word "function". Usually in mathematics, a function is a mapping from one set to another and part of the definition includes specifying the domain.
However, science and engineering types use the phrase "is a function of" to mean "depends on". More precisely, they would say "$y$ is a function of $x$" to mean "if you know $x$, then $y$ is determined." In this context, the answer to your question, "Is $y$ a function of $x$?" is "Yes, because if I know $x$ then I know $y$." And if $x$ is negative, then I know $y$ doesn't exist.
If you allow complex numbers, then the situation is different. If you know $x$ then $y$ could be one of two numbers, and we wouldn't say that $y$ is a function of $x$.
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