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Solfe
2010-Sep-09, 01:18 AM
I am taking calculus and I seem to have this bewildering problem.

Last week we were discussing range and domains. I look at these things and draw a total blank, I have no idea how to come up with the answer. I know what "an answer" in general looks like for domain and range.

Is it so simple as to generate a graph from a statement, look at it and decide what those values are?

My teacher has taken the time to explain it to me twice and I am not ready to go back for a third lesson. She seems honestly baffled with my question because I am otherwise doing fine in class.

What class level is domain and range taught at: Algebra, Calculus, Precalc, Trig?

Can someone give me a quick lesson or point me to a very basic website? I survived the first week fine and we are not working on those items now, but I will have a review test and I can totally see several questions on domain and range.

Solfe

JoeNJ
2010-Sep-09, 04:58 AM
Domain is the set of valid inputs to a function.


For example, in the function f(x) = 2x, you can use any value you like for x, and you won't run into any problems - in this case, the domain is the set of all real numbers.

However, if the function were, say, f(y) = 1/y, you probably see that you would not be able to come up with an answer if y were 0. In this case, your domain would be the set all real numbers except 0.

Similarly, if the function was f(z) = sqrt(z), if z were a negative number, you again will run into problems finding a non-imaginary answer to the function, thus the domain is all non-negative real numbers.

Range is the set of possible outputs from a function.

Using f(x) = 2x, what sort of outputs can be expected if you plug any number in its domain into the function? If we use any real number for x, we can only get a real number out - and since there are no places where output "stops", the range consists of all real numbers.

Using f(y) = 1/y, where is the gap in outputs? Since we know the domain is all real numbers except 0, and that by using any real number except 0 in this function we will receive a real number, the range must be all real numbers except 1/0 - I don't remember the notation for range exactly, but it's something like (-infinity, 1/0], [1/0, infinity), noting the negative on the first infinity.

Suppose the function for which you want to find the range is f(a) = a^2. (squared) If we use a positive number for a, we still get a positive number as an answer. Zero poses no problem for this function either, as 0 is a perfectly legitimate answer. Now what happens if we use a negative number for a? The result is still positive! So in this case, we will never get a non-negative result from the function. So here, the range is all non-negative real numbers.
Try finding the domain and range for these:

f(x) = x^4 - 50

f(y) = x^3 + 9

f(z) = -(z^2)
If you're still lost, try reading this page: http://mathforum.org/library/drmath/view/54551.html

Ken G
2010-Sep-11, 06:30 PM
Another way that might help you get domain and range is to draw perpendicular x-y axes, and then draw some arbitrary squiggle inside the plot. The squiggle generates a mapping from points on the x axis to points on the y axis, called a "function", by drawing lines parallel to the y axis emanating from the x points, until they intersect the squiggle, and then draw a line parallel to the x axis from the intersection points to the y axis. Now imagine taking a flashlight and shining it parallel to the y axis onto the squiggle so that it casts a shadow of the squiggle onto the x axis. That shadow is the "domain" of the squiggle, because any point in that shadow can be mapped onto the y axis using the squiggle. Similarly, the shadow of the squiggle on the y axis from a light shining parallel to the x axis is the "range" of the squiggle, because every point in that shadow is a place where some x point will go.

A wrinkle then emerges-- some squiggles will double back on themselves such that a line parallel to the y axis emanating from a point on the x axis might intersect with the squiggle in several places, yielding several y values associated with that x value. That is called a "one to many" kind of function, which is generally regarded as an invalid type of function. No matter, just break the squiggle up into several squiggles until each one is a proper function, with its own shadows on the axes, so its own "domain" and "range." Even then, there could be several x points that map into the same y point by using the squiggles to generate a function. These are called "many to one" functions, which present no problems until you try to invert the function into a one to many function from the y axis back to the x axis. That is resolved by further subdividing the squiggles into many functions that are all one-to-one, so they are both proper functions and invertible into proper functions going the other way. At any point in this process, you still have those shadows-- so you still have a domain and a range to any of the squiggles.

RGClark
2010-Sep-13, 07:31 AM
I am taking calculus and I seem to have this bewildering problem.
Last week we were discussing range and domains. I look at these things and draw a total blank, I have no idea how to come up with the answer. I know what "an answer" in general looks like for domain and range.
Is it so simple as to generate a graph from a statement, look at it and decide what those values are?
My teacher has taken the time to explain it to me twice and I am not ready to go back for a third lesson. She seems honestly baffled with my question because I am otherwise doing fine in class.
What class level is domain and range taught at: Algebra, Calculus, Precalc, Trig?
Can someone give me a quick lesson or point me to a very basic website? I survived the first week fine and we are not working on those items now, but I will have a review test and I can totally see several questions on domain and range.
Solfe

Your title implies the answer of what course this would be first discussed, in Math 101 class, usually the first math class you would take in university unless you tested out of it by placement exams.
The textbook they use for Math 101 in your school, or whatever the name of the course for the general math class given to most entering freshman, should have a good discussion of the concepts.

Bob Clark

RGClark
2010-Sep-13, 07:34 AM
...

Using f(y) = 1/y, where is the gap in outputs? Since we know the domain is all real numbers except 0, and that by using any real number except 0 in this function we will receive a real number, the range must be all real numbers except 1/0 - I don't remember the notation for range exactly, but it's something like (-infinity, 1/0], [1/0, infinity), noting the negative on the first infinity.
...

What do you mean by 1/0?

Bob clark

HenrikOlsen
2010-Sep-15, 02:20 PM
I think he meant to say something like: ]-inf ... 0[, ]0 ... +inf[ as in all real numbers from (but not including) -infinity to (but not including) 0, and from (but not including) 0 to (but not including) +infinity.

tashirosgt
2010-Sep-15, 09:43 PM
It's interesting that mathematics is often advocated as way to teach logic and precision, but much of it actually focuses on teaching cultural conventions For example if a problem "gives" you the function f(x) = 1/x, it hasn't really given you a function because a function isn't fully defined until its domain and range are stated. The cultural convention in a course whose math takes place on the real numbers is that you are supposed to assume a function "given" by an algebraic expression uses all real numbers for which the expression can produce a defined result and which are not specifically excluded by some other restriction "given" to you (such as "x > 4" etc.). Much of the content of secondary math texts can be explained as social conventions that make it easy for the author to phrase the exercises in the book.

Solfe
2010-Sep-18, 12:14 AM
Thanks guys! I try to log in once week, but with these classes I am doing it less and less. I am crash studying this weekend and I should print this thread.

The book I am using is Calculus Early Transcendentals; Single Variable, 9th edition by Howard Anton. I have to say this edition is very light on anything of mathematical importance. This isn't a slam, I own 8th edition and it is clearly the superior book, which is odd because it has the same author. Yeah ebay! Someone at school recommended it for its clarity and it helps a lot.

I landed in calculus due to my placement test. Its the first math class in the Engineering Science curriculum, but for 3/4 of my class it is the last math class they need to take. Most of them had a choice between calculus and statistics and a few are cursing this class as a poor choice. Personally, I believe that statistics is tougher than Calc 1.

Back to JoeNj - the domain is all real numbers for each; f(y) = x^3 + 9 has a range of [9, infinity) and f(x) = x^4 - 50's range is [-50, infinity) and the last has a range of all reals. I am making headway, assuming those answers are correct.

Thank you all again,

Andrew D
2010-Sep-18, 06:16 PM
Domain is the set of all possible values for x. Range is the set of all possible values for y.

For example, for the function y=sinx, the domain is all real numbers, and the range is all values of y between -1 and 1, inclusive. For the function y=sqrt(x)+1, the domain is all values of x greater than or equal to 0, and the range is all values of y greater than or equal to 1. Graph these functions, and observe why there are so.

There are two methods with which to note domain and range:

"Interval notation" displays the lower limit, then a comma, then the upper limit. Inclusion or exclusion of the limits in the set is noted using parenthesis or brackets before the lower limit and after the upper limit. If the set is not continuous, a unity symbol "U" connects two intervals into a single set. For all real numbers except 0 (the domain of y=1/x) would be: (-infinity,0) U (0,infinity). Hard brackets denote inclusion, and parenthesis denote exclusion; for example "y is less than 4 or greater than 6" would be "(-infinity,4) U (6,infinity)", while "y is less than or equal to 4 or greater than or equal to 6" would be (-infinity,4] U [6,infinity). "Y is greater than 4 and less than or equal to 6" would be (4,6].

Set notation defines the possible values variable as a set, usually "D" for domain and "R" for range. In set notation the an example from above would be:

"Y is greater than 4 and less than or equal to 6" would be D={y|4 lessthan y lessthan-or-equal-to 6}, which would be read as "the domain is equal to the set of all y such that y is greater than 4 and less than or equal to 6.

JoeNJ
2010-Sep-22, 10:17 PM
Back to JoeNj - the domain is all real numbers for each; f(y) = x^3 + 9 has a range of [9, infinity) and f(x) = x^4 - 50's range is [-50, infinity) and the last has a range of all reals. I am making headway, assuming those answers are correct. Thank you all again,

You are correct on the domains and f(x)'s range. However, for f(y), remember that an odd exponent can still allow a negative number in the result. For f(z), note that the expression in the parentheses is z^2, which can only give a positive output. However, the negative outside the parentheses negates z^2, which means that function can only output non-positive numbers.