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Do each of these equations make $y$ a function of $x\text{?}$

Solution

The equation $5x^2-4y=12$ does make $y$ a function of $x\text{.}$ You can isolate $y$ in terms of $x\text{.}$ A few steps of algebra can turn $5x^2-4y=12$ into $y=\frac{5x^2-12}{4}\text{.}$ Now you have an explicit formula for $y$ in terms of $x\text{,}$ so $y$ is a function of $x\text{.}$

The equation $5x-4y^2=12$ does not make $y$ a function of $x\text{.}$ You cannot isolate $y$ in terms of $x\text{.}$ You might get started and use algebra to convert $5x-4y^2=12$ into $y^2=\frac{5x-12}{4}\text{.}$ But what now? The best you can do is acknowledge that $y$ is either the positive or the negative square root of $\frac{5x-12}{4}\text{.}$ You might write $y=\pm\sqrt{\frac{5x-12}{4}}\text{.}$ But now for almost any valid $x$ -value, there are two associated $y$ -values.

The equation $x=\sqrt{y}$ does make $y$ a function of $x\text{.}$ If you try substituting a non-negative $x$ -value, then you can square both sides and you know exactly what the value of $y$ is.
If you try substituting a negative $x$ -value, then you are saying that $\sqrt{y}$ is negative, which is impossible. So for negative $x\text{,}$ there are no $y$ -values. This is not a problem for the equation giving you a function. This just means that the domain of that function does not include negative numbers. Its domain would be $[0,\infty)\text{.}$