Question

Describe the difference between the explicit form of a function of two variables \(x\) and \(y\) and the implicit form. Give an example of each.

Short answer

In explicit functions, the dependent variable is expressed directly in terms of independent variables, like \(y=f(x)=2x+3\), while in implicit functions, both variables are intermingled not separated clearly, like \(x^2+y^2-4=0\).

Step-by-step solution

Definition of an Explicit Function

An explicit function is a type of function where one variable is expressed entirely in terms of other variables. Typically, the dependent variable is written as a function of the independent variables. An example of an explicit function is \(y=f(x)=2x+3\). In this case, \(y\) is expressed explicitly in terms of \(x\).

Definition of an Implicit Function

An implicit function is one where the dependent and independent variables are intermingled and not clearly separated. Rather than solving for one variable in terms of the other, an equation is given that includes both variables. An example of an implicit function is \(F(x,y)=0\), such as \(x^2+y^2-4=0\). In this equation, neither \(x\) nor \(y\) is expressed solely in terms of the other variable.

Highlighting the Difference

The key difference between explicit and implicit functions lies in the way the variables are expressed. In explicit functions, the dependent variable is clearly written as a direct function of independent variables. In implicit functions, the functional relationship is not as directly stated, and both variables are intertwined in the equation.