Domain, Range, and Translating Functions

Date: 06/29/2004 at 22:10:38
From: Scarlet
Subject: Functions, domain, range, absolute value etc. - Algebra 

I'm doing my summer review packet for my Algebra 2 With Analysis class 
(for next school year), but I'm stuck on a function question.
 
1. Given the graph of y = f(x), domain [-4,3], range [-2,3], I am then 
asked to sketch four other functions, such as y = |f(x)| and y = 
f(x - 2) and find the domain and range of each.

I've sort of figured out that y = f(x - 2) should move the graph to 
the right by 2 units, and y = (1/2)f(x) should make the graph thinner
(I don't know how to describe it).  But how do I determine the domain
and range of each?

Oh, the original graph of f(x) is sort of like a V--it's two line
segments, but they don't have the same steepness or slope.

Date: 06/29/2004 at 23:25:05
From: Doctor Peterson
Subject: Re: Functions, domain, range, absolute value etc. - Algebra 

Hi, Scarlet.

Here are some pages on similar problems:

  Graph with f(x)
    http://mathforum.org/library/drmath/view/54509.html 

  Translating Functions
    http://mathforum.org/library/drmath/view/54472.html 

  Graphing Absolute Values
    http://mathforum.org/library/drmath/view/60968.html 

  Graphing the Absolute Value/Square Root of a Function
    http://mathforum.org/library/drmath/view/61112.html 

Those show how to graph a transformed function.  Once you've done 
that, you can use the features you graphed to determine the domain and 
range.  Based on your description, let's suppose the original graph 
looks like this:

                |
    *           +
     \          |
      \         +
       \        |
        \       +        *
         \      |       /
  --+--+--\--+--+--+--/--+--
           \    |   /
            \   + /
             \  /
              * +
                |

The domain is the "shadow" under it; that is, it's the set of values 
on the x-axis for which the function is defined.  Visually,

                |
    *           +
    :\          |
    : \         +
    :  \        |
    :   \       +        *
    :    \      |       /:
  --+--+--\--+--+--+--/--+--
           \    |   /
            \   + /
             \  /
              * +
                |

    +====================+
   -4       domain       3

The range is the "shadow" on the y-axis, that is, the set of values of 
y that the function ever takes.  Look for the lowest and highest 
values, as well as any possible gaps:

                |
    *. . . . . .+              + 3
     \          |              |
      \         +              |
       \        |              |
        \       +        *     |
         \      |       /      |range
  --+--+--\--+--+--+--/--+--   |
           \    |   /          |
            \   + /            |
             \  /              |
              *.+              +-2
                |

Now, if you transform a function by shifting it right, the domain is 
shifted right.  If you transform it by shifting it up, the range is 
shifted up.  For other cases, you can figure out how the domain and/or 
range are affected by thinking about how the graph is changed.  In 
some cases you have to do some extra thinking, such as determining 
exactly where certain points on the graph are transformed to.

Here are some pages on domain and range:

  Range and Domain of a Graph
    http://mathforum.org/library/drmath/view/54576.html 

  Domain and Range
    http://mathforum.org/library/drmath/view/54551.html 

  Interval Notation
    http://mathforum.org/library/drmath/view/52929.html 

If you have any further questions, feel free to write back.

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

Date: 07/01/2004 at 15:35:06
From: Scarlet
Subject: Thank you (Functions, domain, range, absolute value etc. -
Algebra )

Dear Dr. Peterson,

THANKS A LOT for your help!  Your explanations have been very useful
and easy to understand--not just for me but for some of my friends who 
had the same or similar questions as well!  It was especially helpful 
as many of my teachers are not available for help during the summer.  
THANKS AGAIN for all your help!!

- Scarlet