Implicit FunctionsDate: 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/ |
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