A function associates one and only one y-value with any given input value.
y = 4x+13 has a straight line graph; if you draw a vertical line through this straight line graph, your vertical line will intersect the straight line graph in only once place. Thus, y=4x+13 represents a function. The fact that x has the first power here is also a clue that this represents a function.
x^2 + y^2 = 16 is the graph of a circle with center (0,0) and radius 4. Note that we can solve this for y:
y^2 = 16 - x^2, or y = plus or minus sqrt(16-x^2). Thus, for ONE x value there are TWO associated y values. This is a sure sign that this relation is NOT a function.
Similarly: y^2 = (1/3)x + 6 is NOT a function, because when y^2 = (1/3)x + 6 is solved for y, there will be TWO values.
