Implicit relationship

Summary

• Given two variables \(x\) and \(y\) , if \(y=f\left( x \right)\) for some function \(f\) then w e say that there is an explicit relationship between the variables where \(x\) determines \(y\) . Example: \(y=x^2\) .
• A relationship between two variables described by an equation is called an implicit relationship between the variables. Example: \(x^2+y^2=1\) .
• Given an implicit relationship between two variables and given a specific value for one of them, the value / values of the other variable can be determined by solving the equation.
• Example: \(x^2+y^2=1\) and \(x=0\) . Then \(y=1\) or \(y=-1\) .
• Given an implicit relationship between two variables, we say that each variable is implicitly determined by the other .
• In some cases, an implicit relationship can be turned into an explicit one where one variable is determined as a function of the other.
• Example: the implicit relationship \(2x+3y=6\) can be written as an explicit relationship where \(y\) is a function of \(x\)

\[y=2+ \frac{2x}{3}\]

• \(2x+3y=6\) can also be written as an explicit relationship where \(x\) is a function of \(y\) :

\[x=3+ \frac{3y}{2}\]

• In some cases, an implicit relationship cannot be turned into an explicit function.
• Example: The implicit relationship \(x^2+y^2=1\) cannot be turned into a function, \(y=f\left( x \right)\) since there are two \(y\) -values for each \(x\) when \(-1<x<1\) .
• Implicit relationships are more general than explicit ones as they do not need to satisfy the vertical line test.