A formula is a special type of equation. A formula is an equation in which the function name, or dependent variable, has been isolated on one side - “solved for”.

Given an equation for a function, we prefer to work with an equivalent equation where the function has been isolated. However, that is not always possible.

In these situations, it is quite common for the equation to allow multiple values for the dependent variable, which violates our one and only rule for functions. Solving for the function requires choices from these multiple values.

But, this might be ok. We might just be examining the prospective function around a particular domain number. Restricting the algebraic description provided by the equation to that area might clear up the problem.

Functions defined via these types of restrictions are called implicitly defined functions or implicit functions.

If you can isolate the dependent variable and obtain a formula for the function, then we use the word explicit.

Learning Outcomes

In this section, students will

• analyze functions via equations.