Hyperbola question

Question

the graph $ y^2=16x $ is a hyperbola; it can be rewritten as $ y= \pm 4\sqrt{x}$ when I draw it down however It is clearly not a function..question is whether it has to be one in order to perform standard operations on it? (say integration etc...)

Answer 1

The graph of $y^2 = 16 x$ is a "horizontal" parabola opening to the right, not at all a hyperbola.

Technically speaking, if we are taking $y$ as a "function" of $x$ then you are correct: it does not define a function.

But it does define a function if we take $x$ to be a function of $y$. And note that we can integrate (and differentiate) with respect to $y$.

Answer 2

If you re-write the equation to $\displaystyle x = \frac{y^2}{16}$, and treat $x$ as a function of $y$, you can perform functional operations on it, such as differentiation, integration etc.