1 Implicit and explicit functions
Equations such as , , are said to define explicitly as a function of because the variable appears alone on one side of the equation.
The equation
is not of the form but can be put into this form by simple algebra.
Task!
Write as the subject of
We have so
We say that is defined implicitly as a function of by means of , the actual function being given explicitly as
We note than an equation relating and can implicitly define more than one function of .
For example, if we solve
we obtain so defines implicitly two functions
Task!
Sketch the graphs of
(The equation should give you the clue.)
Since is the well-known equation of the circle with centre at the origin and radius 1, it follows that the graphs of and are the upper and lower halves of this circle.
Sometimes it is difficult or even impossible to solve an equation in and to obtain explicitly in terms of .
Examples where explicit expressions for cannot be obtained are