A function assigns to every x-value in its domain unique y-value.
is not a function. To see this, we should first determine for which values of x the equation is even defined. Since
y²≥0, we must have 3x²-4x-8≥0.
You can solve this inequality with the quadratic formula and will find that the allowed values for x are
(-∞,-(2/3)√7+(2/3)] ∪ [(2/3)√7+(2/3),∞)
Let's take one of those allowed x-values, let's say x=10, and see what happens:
Now y can be +√252 or -√252, so there is no unique y-value for x=10, so y is not a function of x.
More generally, when you solve for y,
so you always get two y-values - not allowed for a function!
You can also see that your equation is not a function when you graph it. You will get a hyperbola with two branches, which does not pass the so-called vertical line test: There are vertical lines that will intersect either branch of the hyperbola twice, which cannot happen for the graph of a function.