# 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.

5 Oct 2010 at 9:22 am [Comment permalink]

This is really cool!. Your blog really helps me a lot. I love the t-shirt as well. Could I know where you found it?

5 Oct 2010 at 2:30 pm [Comment permalink]

WA can do it too π

http://www.wolframalpha.com

and then type:

plot (x^2 + y^2 - 1)^3 = x^2 y^3

5 Oct 2010 at 4:04 pm [Comment permalink]

Yes! The graph in the post comes from Wolfram|Alpha.

7 Oct 2010 at 4:35 am [Comment permalink]

So marvelous explanation about the t shirt especially I am a math teacher

Thanks.

7 Oct 2010 at 6:21 pm [Comment permalink]

superb...nicely explained..!!!!!!!!!!!

8 Oct 2010 at 10:57 pm [Comment permalink]

I like it!! To draw this graph i often use MATLAB.

13 Oct 2010 at 6:34 pm [Comment permalink]

i realy liked the way you express you maths. Keep on doing that.

16 Oct 2010 at 12:20 pm [Comment permalink]

If you have a ti-89 you can graph the implicit 2d function in the 3d mode by typing in the equation and formatting the plot to make it graph implicitly. The window will have to be seriously adjusted though.

20 Oct 2010 at 2:13 am [Comment permalink]

your t-shirt is interesting

23 Oct 2010 at 3:08 pm [Comment permalink]

oh thank you I have been asked directly to Bagatrix and not precalculus graph sholved but must be solved to complete the curves and graphs, again thank you ...

22 Nov 2010 at 12:55 pm [Comment permalink]

wow!! amazing.. I like it..

thanks! i think i should suggest it to our math club as a batch shirt..

i like the graphing of the equation in 3-D.