A function is a process or a relation that associates each element x of a set X to a single element y of another set Y.
A. The equation [tex]x=5[/tex] represents vertical line. This is not a function, because one element x=5 is associated to all (not one) elements of [tex]y\in (-\infty,\infty).[/tex]
B. The equation [tex]x=y^2+9[/tex] represents parabola that goes in positive x-direction. This is not a function, because each [tex]x>9[/tex] is associated to two different y. For examlpe, when x=10, then [tex]y_1=1,\ y_2=-1.[/tex]
C. The equation [tex]y=x^2[/tex] represents parabola that goes in positive y-direction. This is the function, because for each x is associated only one y.
D. The equation [tex]x^2=y^2+16[/tex] represents hyperbola. This is not a function, because for one x there are two different possible y.
Answer: correct choice is C
