Algebra

Date: 5/10/96 at 10:26:55
From: Anonymous
Subject: Algebra

Find the domain, range, and zeros of

     g(x) = x^2 - 6x + 4

Date: 10/17/96 at 12:51:17
From: Doctor Leigh
Subject: Re: Algebra

The domain of a function is all the values of x that can be put into 
the function. In this case the domain is all real numbers.

The range of the function is all the values of Y that the function 
covers. This function has a range of y >= -5.

The zeros of a function are found by setting the function equal to 
zero and solving for x. The zeros of this function are 

     3 + sqrt(5)   and   3 - sqrt(5)

Use the quadratic equation.

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

Date: 09/01/2000 at 17:58:48
From: Tim Greene
Subject: Algebra

Hello Dr. Math:

I would like to expand upon the solution given for a problem in your 
archives. In this problem, the writer states that the range of the 
function x^2 - 6x + 4 is y >= -5. No method is shown for obtaining this 
result. I assume the writer determined the function had a minimum value 
when x = 3 and substituted that value of x in the function to find the 
lower limit of the range for the function.

Of course calculus shows us quickly that the minimum value is when x = 3; 
however, many high school students will not know calculus. The value of x 
for which the function has a minimum, and that minimum value, can both be 
obtained by completing the square in the given function; most second year 
algebra students and many first year algebra students will be able to 
understand this approach.

     y = x^2 is a parabola opening upwards with vertex at (0,0);

     y = x^2 + k is a parabola opening upwards with vertex at (0,k); and

     y = (x-h)^2 + k is a parabola opening upwards with vertex at (h,k)

Therefore, 

     g(x) = x^2 - 6x + 4

          = (x^2 - 6x + 9) - 5

          = (x-3)^2 - 5

is a parabola with vertex (3,-5), so the range of the function is 

     [-5,infintiy]   or   y >= -5.

Tim Greene