# I heart math

By Murray Bourne, 04 Oct 2010

I saw this T-shirt recently.

What does it mean and what's that equation?

This is an example of an **implicit function**. When we first learn about functions, they are written explicitly, for example:

*f*(*x*) = sin(*x*) + 4*x*

This explicit function involves one dependent variable only and for each *x* value, we only get one *f*(*x*) value. Of course, I could also write this as

*y* = sin(*x*) + 4*x*

Notice *y* is on the left by itself, and the terms involving *x* are on the right, by themselves.

But there are many functions that are really messy when written explicitly, and so we turn to **implicit** functions.

In implicit functions, we see *x*'s and *y*'s multiplied and mixed together.

## A simple example

A simple example of an implicit function is the familiar equation of a circle:

*x*^{2} + *y*^{2} = 16

In this simple case, we can turn this into an **explicit** function by solving for *y* and getting 2 solutions:

or

But often it is very difficult, if not impossible, to solve an implicit function for *y*.

## The t-shirt Function

Returning to the t-shirt example, we have the implicit function:

(*x*^{2} + *y*^{2} − 1)^{3} = *x*^{2}*y*^{3}

We can expect more than one *y*-value for each *x*-value.

To graph it, we proceed as follows. Let's choose some easy values of *x* and *y*.

If *x* = 0, we substitute and obtain:

((0)^{2} + *y*^{2} − 1)^{3} = (0)^{2}*y*^{3}

(*y*^{2} − 1)^{3} = 0

We get 2 solutions, *y* = ± 1.

Now, let* y* = 0, and we get:

(*x*^{2} + (0)^{2} − 1)^{3} = *x*^{2}(0)^{3}

(*x*^{2} − 1)^{3} = 0

This gives us 2 solutions, *x* = ± 1.

So we know the curve passes through (-1, 0), (0, -1), (1, 0) and (0, -1),

Now, we choose some values of *x* between 0 and 1. We start with *x* = 0.2:

((0.2)^{2} + *y*^{2} − 1)^{3} = (0.2)^{2}*y*^{3}

This gives:

(-0.96 + y^{2})^{3} = 0.04y^{3}

Solving this for *y* gives the real solutions: *y* = -0.824 or *y* = 1.166 (and 4 complex solutions).

We choose some more values and construct a table containing the real solutions:

x |
0 | 0.2 | 0.4 | 0.6 | 0.8 | 1 | 1.2 |
---|---|---|---|---|---|---|---|

y_{1} |
-1 | -0.824 | -0.684 | -0.520 | -0.307 | 0 | complex |

y_{2} |
1 | 1.166 | 1. 227 | 1. 231 | 1.170 | complex |

This equation is symmetrical, so we get the same correspnding values for -0.2, -0.4, -0.6, -0.8, -1 and -1.2.

In fact, outside of this range of *x*-values, there are no real *y*-values.

If we take a lot of points and join them, we get the following graph:

So the t-shirt means "I heart math" (that is, "I love math").

## 3-D Example

Here's another one in 3 dimensions. The implicit function is:

for -3 ≤ *x, y, z* ≤ 3 (which means each of *x*, *y* and *z* takes values only between -3 and 3).

And here's the shirt:

Learn more about implicit functions:

Differentiation of implicit functions

See the 11 Comments below.