How can you prove algebraic inputs of a transcendental function are transcendental?

Question

In general, it's not true that a given transcendental function $f(z)$ will give you a transcendental output for countably infinite algebraic inputs, but is there a condition that can prove that property is or isn't true for a specific function? The proofs that $e^{a}$ is transcendental are complicated and unique, but what about other transcendental functions? Isn't there something that can generalize the results by this point?