If, in a function the dependent variable y can be explicitly written in terms of independent variable x i.e. terms of 'x' must not involve y in any manner then the function is called an explicit function e.g.

y = x² + 1

y = sin x + cos x

If the dependent variable y and independent variable x are so convoluted in an equation that y cannot be written explicitly as function of x then f(x) is said to be an implicit function.

e.g. x² + y² = tan^-1 xy.