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Fri, Jul 13 2007 400 Functions The blending function is that which takes an apple
and turns it into apple juice.
You can also use the blending function on strawberries
and get strawberry juice.
But you can't blend oranges.

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For clarity, we will call this function "blended()".

We use parenthesis, (brackets),
to show that a function acted on something
by writing like this:


and then we can write down the result:

blended(apple) = appleJuice


blended(x) = xJuice

but only if x is in the universe where blended() operates.

the function:
double(x) = 2x
can be drawn easily on paper:

Make a horizontal line from one side of the paper to the other.
Divide it into 21 parts by drawing short vertical lines
across the line, and number them from -10 to 10.
Lets call this drawing "The X axis".
The Y axis is vertical, and is otherwise the same.
Once you draw it, we call the drawing, "The Axis System".

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If X is 5, than double(x) is 10,
so we draw a dot above the 5 on the X axis, (where x=5),
at the height where y=10.

double(-6) = -12 and is not on the paper,
so we draw all the dots from x equals -5 to 5,
and connect the dots with a ruler.

Together with the axes, we call this drawing now:
"The graph of y = 2x".

What would you blend to get absolutely nothing?

Well, if you put nothing in, you'll get nothing back,
that's for sure.

But is it that obvious that you cannot put a few ingredients
that would cancel each other out and you'll still get nothing.
For example if you put in the blender
matter and anti matter in just the right quantities?

The root of a function is the value of X where the value of Y is zero.
So at least one root of blended() is also zero.

Finding the root of a function is like solving an equation:

blended(x) = nothing.
What is x?

y = double(x)
what is X if y is zero?
y = 2x
0 = 2x
2x = 0
x = 0

But with the function:
y = 2x - 2
the root is 1.

So what is the root of y = x*x?

y = x*x -4 ?
2, right.

Well if we look carefully, -2 is also a root of this function.

The equation:
square(x) - 4 = 0
Is the equation to find the root of the function:
y = square(x) - 4

If in an function definition x is squared,
but no higher powers are there,
we call the function a Quadratic function,
and the equation a Quadratic equation.

I you draw the line of the function, the roots of the equation
are those where y=0, or when the line crosses the X axis.

We also use letters for the numbers in the definition itself,
and not to be confused with the numbers from the axes,
we take the letters A,B and C for the Constants in the equation,
and X,Y for the variable numbers from the axes
we originally marked X and Y in our drawing of the graph.

y = a*sqaure(x) + b*x + c

The quadratic equation to find the roots of
the quadratic function is:

a*sqaure(x) + b*x + c = 0

If all this down to earth mathematics
was a bit stressful,
remember at least that quadratic is not Quadruped,
nor Aquatic, which is more on the Oceanic side,
that math can predict the future,
that the suns will rise tomorrow,
in a galaxy, far, far, away....

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Keep Up the Good Work.