Implicit Functions

Date: 11/26/97 at 10:05:20
From: Kristen Norris
Subject: Implicit functions

What is an implicit function? Please give me a definition and 
several examples.  Also, please tell me how it is used and give me 
real life related applications.

Date: 11/26/97 at 14:45:23
From: Doctor Rob
Subject: Re: Implicit functions

Whenever you have an equation relating two variables, each is an 
implicit function of the other. Even if you are unable to solve the 
equation explicitly for one of the variables in terms of the other, 
given a value of one variable, there is a value of the other which 
makes the equation true. That defines the second variable as a 
function of the first.

For example, if you knew that y + e^y = x^5, the equation would define 
y as a function of x implicitly. Solving for y is not possible, so you
cannot express the function algebraically, but for each value of x, 
there is a unique value of y which makes it true. You can compute that 
value by numerical methods. For example, when x = 2, the value of y is 
(about) 3.35497961935092. Similarly, x is a function of y, but this 
time it is possible to write the function down explicitly, x being the 
fifth root of y + e^y.

Any time you have a complicated equation involving two variables, this
situation may arise.

In order to be a *function*, of course, given a value of the 
independent variable, there must be a *unique* value of the dependent 
variable which makes the equation true. If there is more than one, 
you don't have a function, but something called a "relation". For the 
sake of clarity, consider x to be the independent variable, and y the 
dependent variable.  For x = y^2, y is *not* a function of x, because 
for each positive value of x, there are two values of y which work:  
y = Sqrt[x] and y = -Sqrt[x].

This means that for y to be an implicit function of x, if you graph 
the solution set of the equation, every vertical line x = a should 
intersect the graph in a unique point. Then the value of the function 
y at x = a is the y-coordinate of that intersection point.

One simple case where an implicit function always exists is if the 
graph is monotone increasing, that is, if (a,c) and (b,d) are points 
on the graph, and a > b, then c > d. Another case is when it is 
monotone decreasing (a > b ==> c < d). Implicit functions can also 
exist in other situations, however. The example y + e^y = x^5 has a 
monotone increasing graph, and that is why I knew that y was an 
implicit function of x.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/