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/
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