An implicit function is a function specified by a relation between the input and output that is left unsolved
for the output. An equation 
defines 
on a domain 
if it determines a unique allowed value of 
for each 
. The choice of allowed values
can select a branch. For example, the circle 
defines the function 
on ![[-1,1]](/images/equations/ImplicitFunction/Inline8.svg)
when restricted to 
, and the function 
on the same domain
when restricted to 
.
An example in which a single equation already determines a unique output is Kepler's equation. For a body orbiting on an ellipse of fixed
eccentricity 
, the mean anomaly
is known from the elapsed time, and the eccentric
anomaly 
is found from
 |
The right-hand side has derivative 
and tends to 
as 
. Although the equation gives 
explicitly in terms of 
, it defines 
implicitly as a function of 
, with exactly one real 
for each real 
.
