Graphing Two Functions With One Equation

Date: 10/21/2003 at 22:02:55
From: vivian
Subject: Is it possible to use ONE equation to graph two functions

Hi Dr. Math. I have a question on graphing functions.  I am just 
wondering if it is possible to use one equation to graph two 
functions?  For example, is there one equation that exists that can 
be graphed into:
         _
1. y = \/x - 4
         _
2. y = \/x - 8

I am just wondering, as I am doing my math homework...

Date: 10/21/2003 at 22:42:13
From: Doctor Peterson
Subject: Re: Is it possible to use ONE equation to graph two functions

Hi, Vivian.

Interesting question!

Obviously you can't have an equation of the form

  y = f(x)

represent two functions of x, since it can only have one y value for 
each x. But an equation like

  f(x,y) = 0

can easily have two values of y for each x. For example, the equation 
of a circle represents two functions.

In fact, let's suppose we have any two functions, f and g. We want to 
make an equation such that (x,y) satisfies it whenever either

  y = f(x)  or  y = g(x)

Can we do that? I've never tried before, but I have an idea. In fact, 
two ideas. First, we can use the zero-product property:

  (y - f(x))(y - g(x)) = 0

will be true under exactly the right conditions.

Second, we can use the fact that

  |y| = h(x)

is true when y is either h(x) or -h(x). So if we take the average of 
our two functions, [f(x) + g(x)]/2, then we have to add either the 
positive or negative of the same quantity to get either function, and 
this equation represents the union of both functions:

  |y - [f(x) + g(x)]/2| = [f(x) - g(x)]/2

Think about that and you should see why it works. It's not nearly as 
nice as the first way, but maybe that makes it more fun!

Can you apply this to a specific pair of functions like yours and get 
an equation to graph? Try simplifying each equation; one way in the 
second case is to square the equation to avoid having an absolute 
value.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/