



Up to
now, we have seen a lot of equations that look like this:






Y
= (stuff with X's in it)






Now we
are going to give Y another name. 


It's new
name is: 


F(X)






This is
pronounced "F of X" 


It's
really just another name for Y. 





What it
means is ... 


(just
skip this part if you don't care what it means) 


Remember
when we said that in these equations 


X could
be pretty much whatever it wants to be 


and that
Y depended on X. 


That made
Y the dependent variable 


and X the
independent variable. 





Another
way to say that "depended on" thing is 


"is
a function of" 


So: 


Y depends
on X 


Y is a
function of X 


Y is F of
X 


Y is F(X) 


Y = F(X) 


It's sort
of like the evolution of math ... 





OK,
that's it for "What does F(X) mean?" 


You
should come back now ... 





So
instead of 


Y = 3X^{2}
+ 2






You could
say ... 


F(X) = 3X^{2}
+ 2






This name
for Y is like a special club. 


Not just
an old Y = (stuff) equation 


can use
this new name for Y. 


To join
the club, the equation must pass a test. 


The test
is called the vertical line test. 


Here's
how it works. 





First
draw the graph of the equation you're testing. 


We'll try
Y = ^{1}/3X
^{2} + 2. 


This is a
"U" that has it's bottom at X = 0, Y = 2. 











Now
here's the test. 


If we can
draw a vertical line 


(vertical
means "goes straight up and down") 


that
crosses the graph line MORE THAN ONCE 


the
equation does not get to join the F(X)
club. 





In this
case, there is no vertical line that we can draw 


that
crosses the equation line more than once. 


So Y = ^{1}/3X
^{2} + 2 gets to use the new name for Y. 





Y = ^{1}/3X^{2}
+ 2






OK fine,
but we could have some really complicated equation 


that
could take forever to graph. 


Even
then, there could be some little place out at X = ten million 


that we
might miss. 


That
point might be the one and only place a vertical line 


hits the
equation more than once. 


How about
some rules that say what kinds of equations 


meet or
fail the test? 





No
problem ... 





When you
have equations with stuff like: 





Y
= 4X^{6}  3X^{4} + 2X^{3}  X^{2} + 5






They will
pass the test. 


They key
is, we just have Y on the left (no exponents on Y) 


and just
X (exponents OK) and numbers on the right. 





If we
have 2Y = stuff with X's in it 


we can
divide both sides by a number (2) and get Y = stuff with X's in it 


so it
passes ... 





If we
have Y + 3 = stuff with X's in it, 


we can
subtract 3 from both sides to make it Y = stuff with X's 


so it
passes. 





BUT ... 





If we
have: 


Y^{2}
= stuff with X's in it 


IT
FAILS






If we
have: 


Y
= stuff with X's in it 


IT
FAILS






If we
have: 


Y
= ±
(stuff with X's in it) 


IT
FAILS






If the
equation gets to use F(X) 


It also
gets to call itself a function. 





copyright 2005 Bruce Kirkpatrick 
