# How do you decide whether the relation x^2-3y =12 defines a function?

Nov 1, 2015

A function is when for every $x$ there is a maximum of only one $y$

#### Explanation:

Let's rewrite the relation by adding $3 y$ to both sides and subtracting $12$ (also on both sides) and then divide by $3$:

${x}^{2} - 3 y = 12 \to {x}^{2} - 12 = 3 y \to y = \frac{1}{3} {x}^{2} - 4$
graph{x^2/3-12 [-28.86, 28.87, -14.43, 14.43]}
This gives one $y$-value for every $x$, so this may be called $y \left(x\right)$, or $y$ as a function of $x$.

Note :
You can not write $x \left(y\right)$, or $x$ as a function of $y$, because for many $y$'s there are two possible $x$'s.