What does it signify to say that the electric potential is a function of x only? More exclusively what does the equipotential diagram (in 3-dimensional space) look like in the case of an electric potential that is a function of x only.

Answer:

An equipotential figure is a picture indicating for each value from a set of values of electric potential that region of space for which all points have that value of electric potential.

When we utter that the electric potential depends on x merely we are saying that all those points in space whose coordinates have one and the similar value of x will have one and the same value of electric potential. In other sense for a given value of x the electric potential will have a certain value no matter what the values of y and z are now the locus (the infinite set) of points in space that all have the same x coordinate form a plane that is parallel to the y-z plane and is in fact a distance x from the y-z plane. Ever since all these points have one and the same value of electric potential the equipotential regions in the case of a potential which is a function of x only are planes parallel to the y-z plane.