Helpful Tools for Calculus, Chapter 1: Functions and Relations
Previous Section: 1.12 Hyperbolic Functions
For the following chapters we will begin discussing equations that might not be considered functions. Recall that for something to be considered a function it has to pass a ‘vertical line test’. That is to say, each x-value can only have one y-value associated with it. In this and the following chapters there will potentially be equations with multiple values associated with each input.
Implicit Equations
Implicit Equations are different than our regular equations in the sense that generally the dependent variable, y, is explicitly stated.
However, as the name suggests, in implicit equations that variable is not isolated, and in many cases it cannot be isolated.
In pretty much all other aspects, an implicit equation is no different than anything else we have studied thus far. The calculate a point on the curve we plug in an x or y value, but there may be more than one answer. To graph these, we should make a chart of as many points as are necessary to understand the shape of the graph. The domain and range are often easily visible from graphs of the function. Solving for them manually requires identifying when a function as a solution from a combination of variables which can be particularly challenging.
Famous Examples
Many implicit equations are relatively famous and have names associated with them. Below are the equations and graphs of a few of them.
Practice Problems
Example 1: Calculate f(2,4) if f(x,y) is defined as 
Solution: We plug in the values corresponding to each variable to get
Example 2: Calculate the y-value(s) for 
Solution: Begin by substituting the values for x and the output in order to solve for y
Example 3: Calculate the y-value(s) for associated with x=1 on the lemniscate curve.
Solution:
Example 4: Identify the largest value of the domain of the Limaҫon of Pascal.
Solution: From the graph we can identify that the maximum value occurs where y=0.
Based on the graph we can conclude that he maximum value of the domain is 4+√22/2 ≈ 6.345
Example 5: Identify the range of the Folium of Descartes.
Solution: Based on the graph we are able to see that both the domain and the range are the set of all real numbers.
Example 6: Graph x2+y2=9
Solution: This gives the equation of a circle whose radius is 3.
Example 7: Graph y2-x=0
Solution: By solving this for y, we get y=±√x
Review
Implicit Equations are equations where the variable is not explicitly identified. These may have multiple outputs for each input.