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For an algebraic function in one variable, as many as the highest power of the variable.
The presence of any term that is not a constant or a multiple of the independent variable. It can be any other power of that variable, or a trigonometric or exponential or any other function.
Neither variable appears anywhere in the equation in a denominator, or raised to any power except the first power.
Use the definition of a function. If, for any value of one variable, there is only a single possible value of the second variable, then the second variable is a function of the first variable. The second variable is often called the "dependent variable". If you can solve an equation explicitly for the dependent variable, then it is a function. If you can NOT solve it for a variable, it may or may not be a function - it turns out that some equations are hard or impossible to solve explicitly for one of the variables.
Dependent variable