Functions: Domain, Range, and Piecewise

Date: 08/31/98 at 21:07:29
From: Anonymous
Subject: Pre-Calculus Honors

What exactly are piecewise functions? What are open and closed points?  
How do you figure out the domain and range of a function without 
graphing it on a calculator? Please help.

Date: 08/31/98 at 22:18:49
From: Doctor Jaffee
Subject: Re: Pre-Calculus Honors

Hi there,

If you have a function in which x is the independent variable and y is 
the dependent variable, the domain is the set of all the x numbers.  
Normally, x can be any number, but if it is inside a radical sign, you 
need to see whether there are values of x that will make y imaginary.  
Those x numbers can't be in the domain. If x is in the denominator of 
a fraction, any value of x that would make the denominator zero must 
be eliminated. If you can solve the equation for x in terms of y, you 
can use the same procedure to determine the range. Now, this is not a 
complete answer, but it addresses the great majority of cases.

If you have a function that is determined by one expression for some 
values of x, and then is determined by another expression for other 
values of x, then that function is what we call "piecewise." For 
example, y = x^2 for x < 3 and y = 2x + 1 for x > or = 3. On the curve 
y = x^2, x can't equal 3, so the end of the curve, (3,9), is open. 
But on the line y = 2x + 1, x can equal 3, so the end of the line, 
(3,7), is closed.

I hope this explanation clarifies the situation somewhat. If you still 
have questions, write back and I or one of the other doctors will try 
to help.

Good luck in your course.

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