Question

Describe the difference between the explicit form of a function and an implicit equation. Give an example of each.

Short answer

Explicit functions are those in which the dependent variable is given in terms of the independent variables, while implicit equations are equations that do not explicitly solve for one variable in terms of the others. For example, \(y=2x+3\) is an explicit function, while \(x^2+y^2=25\) is an implicit function.

Step-by-step solution

Understand Explicit Function

An explicit function is a function in which the dependent variable (generally denoted as \(y\)) is given in terms of the independent variable (normally denoted as \(x\)). In an explicit function, you can isolate \(y\) on one side of the equation easily. For example, the function \(y = 2x + 3\) is an explicit function because \(y\) is expressed solely in terms of \(x\).

Understand Implicit Function

An implicit function is one where it is not possible (or not easy) to express the dependent variable entirely in terms of the independent variable. Implicit functions are typically equations that involve both \(x\) and \(y\) on the same side. For example, the equation \(x^2 + y^2 = 25\) is implicit, as it is not immediately solved for \(y\).