Notes on FUNCTIONS
------------------------Warren Smith-----Math 75--------

A FUNCTION  is a machine which inputs numbers
and spits out a number.

   y = sin(x)   inputs x and spits out y.

   y = MyFunction(a,b,c) = a + b/(c+4.3)   
              inputs a,b,c  and spits out  y.


The DOMAIN of a function  is the set of all inputs you
could possibly feed it.

   y = sin(x)    DOMAIN = all reals x.
               

   y = MyFunction(a,b,c) = a + b/(c+4.3)   
              DOMAIN: all reals a,b,c  except  c = -4.3   is disallowed
               since it would cause division by 0.

We could define  (using a definition with 2 cases)

   y = MyBetterFunction(a,b,c) = a + b/(c+4.3)   if  c is not -4.3,
                               = a + c - b/17    if  c is  -4.3

which would have domain = all reals a,b,c  since it is now defined 
everywhere.  


The RANGE of a function is the set of all outputs it could possibly
produce.

   y = sin(x)    RANGE = all reals y   in  [-1,1].
               
   y = MyFunction(a,b,c) = a + b/(c+4.3)   
              RANGE = all reals y.
        2
   y = x  + 3       RANGE = all reals y  in  [3, infinity).

           2
   y = 1/(x  - 3)   RANGE = reals y with y>0 or with y in (-infinity, -1/3].


PLOTS:
It is a good idea when plotting functions          y=F(x)
to figure out what x's cause divisions by 0   (need to solve for x such that
some divisor = 0)  since they usually lead to infinities and infinite
jumps in the plot.  It is also a good idea to figure out how the plot behaves
for large positive x and large negative x.

I prefer you label the axes of plots (y and x, say, or whatever the
variable names are, and with units, if any) and put arrows on axes pointing in
the positive direction.  Also label the axes with numbers at important
values.

The plot of a FUNCTION  y=F(x)  has at most one y value for each x value.
(In fact it has exactly one y value for each x value in its DOMAIN,
and no y values at other x.)
But the plot of   F(x,y) = constant,   say,  could have many y values
at the same x value.   For example the x-y plot of

             2           2    2
      (x - a)  +  (y - b)  = c

where a,b,c are constants with c>0, is a CIRCLE with center at 
x=a, y=b  and with   radius=c.   (Use Pythagoras.)
This has TWO y values for every x value in the interval  a-c < x < a+c,
one on the upper semicircle, one on the lower.  So this circle is NOT,
and cannot be, the plot of a FUNCTION.  (Also: the circle has NO y values
for x's with  |x-a| > c.)

The DOMAIN of a function y=F(x) is the set of x on the x-axis you get
by projecting the plot vertically to "cast a shadow" on the x-axis.
That shadow is the domain.  The RANGE is the set of y on the y-axis
you get by projecting the plot horizontally to "cast a shadow" on the x-axis.


You can COMPOSE two functions to get a new function.  Like the
+, -, and times and divide symbols, which operate on numbers
to make a new number, there is a new   FUNCTION COMPOSITION SYMBOL o
which operates on functions to make a new function.

For example    H = F o G   is the new function H(x) = F(G(x)).

    sin o cos  =   sin(cos(x)).

    sin o cos o log  =   sin(cos(log(x))).

Example: if you compose   F(x) = x+7  with  G(x) = 3x  you get
   H = F o G     H(x) = F(G(x)) = 3x+7.
But composing in the other order gives the DIFFERENT answer
   H = G o F     H(x) = G(F(x)) = 3(x+7) = 3x+21   !!

Example: if you compose   F(x) = 2x+7  with  G(x) = log(x)  you get
   H = F o G     H(x) = F(G(x)) = 2log(x)+7.
But composing in the other order gives the DIFFERENT answer
   H = G o F     H(x) = G(F(x)) = log(2x+7).