Range of a Function

Date: 01/05/97 at 09:13:29
From: Anonymous
Subject: Functions

Dear Dr. Math,

Please could you tell me how to find the range of a given function?
For example:

The functions f and g are defined over the set of real numbers by
                         
                         f : x --> 3x-5
                         g : x --> e^-2x
    
a) State the range of g


Thank you for your help in this matter.

Regards,
Tracy Pritchard

Date: 01/05/97 at 17:26:26
From: Doctor Pete
Subject: Re: Functions

Hi,

First, as a side note, it is not a common practice to define functions 
as in the above.  For instance, most people would write g as follows:

     g : R --> R,  g(x) = e^(-2x),

where R denotes the set of real numbers.  The first part tells us what 
type of mapping g is.  Generally speaking, if we write:

     f : U --> V

then f is a function that maps elements from the set U to elements in 
the set V.  U is called the domain, and V is called the codomain, or 
range.  It is often not necessary to be completely specific when 
specifying the codomain; for instance, in the function:

     f : R --> R,  f(x) = x^2 + 1

we could have written

     f : R --> R+

where R+ denotes the set of all positive reals.  This is because x^2 
is always greater than or equal to 0 for all real x so there is no way 
that this function can produce a negative number when the domain 
consists of only real numbers.  On the other hand:

     f : C --> C,  f(x) = x^2 + 1

where C is the complex numbers, one cannot make such a restriction on 
the codomain because the domain has been specified to be the set of 
all complex numbers.

With this in mind, we should write your problem as follows:  If

     f : R --> U,  f(x) = 3x-5,
     g : R --> V,  g(x) = e^(-2x)

specify U and V.

Clearly, U and V are subsets of R (though this is not always the 
case).  We may find V informally by graphing it.  However, of 
particular interest is the fact that the function e^x > 0 for all 
real x.  In other words:

     h : R --> R+,  h(x) = e^x

Also, note that:

     k : R --> R,  k(x) = -2x

It is clear that the range of k is the same as its domain (graph it).  
Finally, it should be observed that:

     g = h o k,  or g(x) = h(k(x))

That is, g is the composition of h and k.  We can then make the 
following diagram:

         k
      R --->R
       \    |
        \   |
       g \  | h
          \ |
           \|     
            R+

So we see that the range of g is R+, the set of positive reals.

In general it is not always easy to completely specify the codomain of 
a function.  Also, it is usually necessary to specify the domain in 
order to determine the restrictions on the codomain, as I have 
mentioned above.

-Doctor Pete,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/