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Math 104: Calculus16 chapters | 135 lessons | 11 flashcard sets

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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.

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*.

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.

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

In this case, I've got a kind of loop thing, and my equation actually implies three different functions. I have my first function, let's say *y*=*f(x)*, that's defined over this range of *x* values, I have another function, *y*=*g(x)*, that's defined over these *x* values and I have a third function, *y*=*h(x)*, that's defined over these values of *x*. So one implicit function implies three real functions of *x*.

You can get some really cool-looking graphs with implicit functions, but it's really important to know that you can't write them everywhere with a single function. That is, the equation 1= *xy* + sin(*x*) + cos(*y*) is an implicit function, but I can't write it as *y*=*f(x)* for all values of *x*.

So let's review. **Implicit functions** are equations that have *x* and *y*, but you can't separate them. You can't solve for *y* as some function of *x*. In general, you want to graph these on a computer or calculator, although there are some implicit functions that you should just know, like 1 = *x*^2 + *y*^2 is a unit circle. Implicit functions also 'imply' one or more functions, like in the case of the circle, 1 = *x*^2 + *y*^2 implies that *y*= the square root of (1 - *x*^2) and *y*= - the square root of (1 - *x*^2).

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Math 104: Calculus16 chapters | 135 lessons | 11 flashcard sets

- What is a Function: Basics and Key Terms 7:57
- Graphing Basic Functions 8:01
- Compounding Functions and Graphing Functions of Functions 7:47
- Understanding and Graphing the Inverse Function 7:31
- Polynomial Functions: Properties and Factoring 7:45
- Polynomial Functions: Exponentials and Simplifying 7:45
- Exponentials, Logarithms & the Natural Log 8:36
- Slopes and Tangents on a Graph 10:05
- Equation of a Line Using Point-Slope Formula 9:27
- Horizontal and Vertical Asymptotes 7:47
- Implicit Functions 4:30
- Go to Graphing and Functions

- Go to Continuity

- Go to Series

- Go to Limits

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