I heart math

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) + 4x

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) + 4x

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:

x² + y² = 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:

(x² + y² − 1)³ = x²y³

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 + y² − 1)³ = (0)²y³

(y² − 1)³ = 0

We get 2 solutions, y = ± 1.

Now, let y = 0, and we get:

(x² + (0)² − 1)³ = x²(0)³

(x² − 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 + y² − 1)³ = (0.2)²y³

This gives:

(-0.96 + y²)³ = 0.04y³

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
y1	-1	-0.824	-0.684	-0.520	-0.307	0	complex
y2	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

Curves in polar coordinates

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