Domain/Range of a Function

Date: 01/22/97 at 18:16:21
From: Jennifer
Subject: Domain and Range

How do you find the domain and range when given the function:
 
f(x) = 2x^2 - 3x+1 

I have tried to graph it, but I don't know where the vertex is.

Date: 01/23/97 at 13:22:21
From: Doctor Mike
Subject: Re: Domain and Range

Hi Jennifer,
  
Let's first get the words straight. Domain is the set of x values for 
which the function is defined.  2x^2-3x+1 makes sense for all real 
numbers x, so its domain is the full set of reals. Range is the set of 
all values that f(x) can have. For this we have to look at the graph 
of the function.  Some of this you may have already done.
 
You can start by factoring it like f(x) = (2x-1)*(x-1) from which you 
learn where the function crosses the horizontal axis. [Where?] Because 
the leading coefficient (2) is greater than 0 we know the graph looks 
like an "infinite U", that is it opens upward. So, going from left to 
right: it swoops down from the upper left, cuts the y-axis at the 
point (0,f(0)) = (0,1), crosses the x-axis going downward, then 
bottoms out at some y-value, starts back upward, crosses the x-axis 
again going upward, and then swoops upwards and to the right and goes 
off to infinity. Basically, one of the standard shapes you see for a 
quadratic polynomial graph.  
    
I hope you are now saying "Yes, yes, but WHERE does it bottom out?"
I was hoping you would ask that. I also hope you are in a class where 
some calculus is being discussed, because the x value where the graph 
bottoms out is the x-value which makes the derivative of your function 
zero. One of the early things you often learn in calculus is to find 
the derivatives of polynomials. For this one, the derivative is 
d(x) = 4x-3. The solution to d(x) = 0 is x = 3/4. This value 3/4 is 
WHERE the f(x) function bottoms out. The functional value there,
f(3/4), is the actual minimum y-value attained by the function as its 
graph bottoms out. f(3/4) had better come out to be negative, because 
we know the graph dips below the x-axis, right?
  
So, f(0.75) is the minimum functional value. There is no maximum
functional value because it goes off to infinity upward. You could 
then say that the y-value "range" is [m,infinity) where m=f(0.75), or 
that the y-value "range" is "all real numbers greater than or equal to 
m". Of course, you need to figure out what m is.  
  
If you don't know any calculus, and the explanation I gave above is
not understandable to you, then perhaps you are just supposed to do 
the graph carefully, and "estimate" where it bottoms out, and find the 
bottoming-out value.  Anyway, I hope this helps. 

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

Date: 01/23/97 at 13:36:48
From: Doctor Mike
Subject: Re: Domain and Range

Hello again Jennifer,
  
There is another way to find out, WITHOUT CALCULUS, where the graph 
bottoms out. Your teacher may have told you that the graph of a 
quadratic polynomial is left-to-right symmetric. That is, you draw a 
vertical line through the point on the graph where it bottoms out (or 
tops out if it opens downward), and the left and right sides of that 
line are mirror images of each other. So how this affects your 
problem is that once you have found the 2 roots of your polynomial, 
the average of those 2 x-values must be the place where it bottoms 
out. This should give you the value x = 3/4. You still have to 
evaluate f(3/4) to get the actual minimum value. That minimum value 
you use to express the range of the function.
  
Again, I hope this helps.  

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