Determining if Y is a Fuction of X in an Equation

How can one determine without graphing if an equation represents y as a function of x. For example what would make {16x^2-y^2=0} not a function? And yet {y=sqrt(1-x)} a function. How is this determined?

Are there varieties of ways of determining weather an equation is a function depending what the equation consists of? Whether the equation consists of absolute values, square roots, rational expressions or combinations? Would there be different methods or formulas for determining this based on what the equation consists of? I hope my question is clear, thank you.