- Identify whether a function is given directly (explicit) or needs solving (implicit)
Understanding Explicit and Implicit Functions
In mathematical analysis, the distinction between explicit and implicit functions is pivotal in understanding how variables interact within an equation. This understanding is fundamental when approaching calculus, as it affects how we might differentiate or integrate expressions with respect to a given variable.
Explicit Functions
An explicit function is one where the dependent variable, typically denoted as [latex]y[/latex], is expressed directly in terms of the independent variable [latex]x[/latex]. In simpler terms, [latex]y[/latex] is isolated on one side of the equation. This direct expression allows us to readily compute the value of [latex]y[/latex] for any given value of [latex]x[/latex] without the need for additional manipulation.
explicit functions
An explicit function clearly expresses the dependent variable, such as [latex]y[/latex], in terms of the independent variable, such as [latex]x[/latex]. It takes the form [latex]y=f(x)[/latex], allowing for direct computation of [latex]y[/latex] for any given [latex]x[/latex].
Consider the function:
For this quadratic equation, the value of [latex]y[/latex] is defined explicitly for each [latex]x[/latex], allowing for straightforward evaluation.
Implicit Functions
Conversely, an implicit function is one where the relationship between [latex]y[/latex] and [latex]x[/latex] is implied within an equation. [latex]y[/latex] is not isolated, and the equation must be manipulated to solve for [latex]y[/latex] in terms of [latex]x[/latex] if it’s even possible. Implicit functions often arise in situations where two or more variables maintain a relationship, but one cannot be neatly expressed in terms of the others.
implicit functions
Implicit functions are those in which the relationship between variables is expressed indirectly. The dependent variable is not isolated on one side but is mixed with the independent variable within an equation. Solving for one variable in terms of the others may not be straightforward or sometimes even possible.
The equation below defines a relationship between [latex]x[/latex] and [latex]y[/latex] where neither variable is isolated as the subject of the formula.
[latex]x^3+y^3=6xy[/latex]
To determine if a function is explicit or implicit, look for the dependent variable ([latex]y[/latex]) and assess whether it is written on its own with respect to [latex]x[/latex].
- If [latex]y[/latex] is by itself on one side of the equation, then it’s an explicit function.
- If [latex]y[/latex] is mingled with [latex]x[/latex] and cannot be easily isolated, then it’s an implicit function.
Determine whether each of the following functions is explicit or implicit:
- [latex]y = 3x^2 - 7[/latex]
- [latex]x^2 + y^2 = 16[/latex]
- [latex]y^3 + 3y = x[/latex]
- [latex]e^y = x + y[/latex]
- [latex]\ln(x) + \ln(y) = 1[/latex]