Implicit Functions

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  • 0:06 Circles
  • 1:28 Ovals
  • 1:49 Implicit Functions
  • 2:34 Example of an Implicit…
  • 3:46 Lesson Summary
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Lesson Transcript
Instructor: Eric Garneau
Sometimes inputting a variable into a function 'black box' doesn't yield a simple output. Find out what happens when you can't isolate the dependent variable on one side of the equal sign.


Have you ever thought about circles? I mean really, really thought about circles? Well, the equation for a circle is x^2 + y^2 = 1.

The equation for a circle
Equation for a Circle

For every value of x inside this circle, so between -1 and 1, there are two possible y values: one on the top of the circle and one on the bottom of the circle. That means that for each value of x we get two values of y, so that's not really a function. Let's solve this so we get y equals some function of x. I'm going to subtract x^2 from both sides, and I'm going to take the square root of both sides. I get y= +/- the square root of (1 - x^2). Well that's really two functions: I have one for the top half of the circle, y= the square root of (1 - x^2), and one for the bottom half of the circle, y= - the square root of (1 - x^2). So what's going on here?


Things might be a little clearer if we take a look at the circle's slightly more complicated cousin, the oval. The equation for an oval is 1 = x^2 + y^2 + xy.

The equation for an oval
Equation for an oval

Implicit Functions

This is what I call an implicit function - it depends on both x and y; x and y cannot be separated. I can't write f(x) equals some function of x; it has to be some function of x and y. It's important to note that functions like this aren't really functions, at least in the traditional sense. What do I mean? Let's go back to our circle, x^2 + y^2 = 1. I solved it for y= the square root of (1 - x^2) and y= - the square root of (1 - x^2), so I have two functions that are implicit in our implicit function. Let's take a look at another example.

Example of an Implicit Function

Let's take a look at xy = y^3 + x^3.

Graph for the final implicit function example
Corkscrew Example

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