Ok, so here is the problem:
Quote4. Not all functions in mathematics arise as nice combinations of previously seen functions. Consider the following two functions:
f(x) = { x if x is rational
0 if x is irrational }
g(x) = { x^2 if x is rational
0 if x is irrational }
(a) You can't graph these functions in the same way that you can other functions that we've seen. However, since it's useful to have an idea of what a function looks like, sketch (by hand) the graphs of f(x) and g(x) at the bottom of the page.
Now part (a) seems easy enough. Although I'm not 100% sure I'm correct.
My hand drawn sketch for the first piece (x and x^2) of each piecewise function (f(x) and g(x)) basically looks like a shit load of inclusive (solid) points, since a rational number is any number that can be expressed by dividing two integers. The second piece of each piecewise function, which are the same for both f(x) and g(x) is simply a horizontal line y = 0 with a bunch of inclusive (hollow) points that exclude any rational x values... am I correct?
EDIT: I originally posted all of the parts (a) - (f) in the problem, but removed them because I think once I know if my thinking/graph for part (a) is correct, I can do the other parts...
Thanks.
f(x) should be x for integers and non-repeating, infinite decimals (pi, e, etc...), and 0 for everything else.
g(x) should be x2 for integers and non-repeating, infinite decimals (pi, e, etc...), and 0 for everything else.
Not too challenging, as you will see once you get further into the course.
Also, assume non-repeating, infinite decimals are included. I don't know your teacher.
Quote from: rabbit on February 14, 2006, 09:09 PM
f(x) should be x for integers and non-repeating, infinite decimals (pi, e, etc...), and 0 for everything else.
g(x) should be x2 for integers and non-repeating, infinite decimals (pi, e, etc...), and 0 for everything else.
Since when are Pi and e rational numbers? Also, 10/11 is an example of a repeating infinite decimal that is a rational number..
I said infinite, non-repeating decimals. Pi has an infinite number of digits, but none ever repeat, whereas something like 1/3 is an infinite repeating decimal.
Quote from: rabbit on February 15, 2006, 06:35 PM
I said infinite, non-repeating decimals. Pi has an infinite number of digits, but none ever repeat, whereas something like 1/3 is an infinite repeating decimal.
Um, ok.. Pi isn't a rational number, therefore f(x) != x when x=Pi. Further, you said f(x)=0 for numbers that aren't integers and that aren't non-repeating infinite decimals; 10/11, and 1/3 are both infinite repeating decimals and are rational numbers. Therefore f(x) = x for x = 10/11 or x = 1/3, not 0. Remember, f(x) = x when x is rational.
Also, how do you know that no digits ever repeat in the expansion of Pi? (which is really irrelevant anyways)
But pi is an infinite, non-repeating decimal. I'm using the definition of 'rational number' as a number which can be expressed as a ratio of two integers which has a terminal decimal number equivolent.
I know no digits ever repeat in the expansion of pi because I'm being a douchebag and assuming that they never do, as there is not any evidence of it being so.
Quote from: rabbit on February 16, 2006, 05:39 PM
But pi is an infinite, non-repeating decimal. I'm using the definition of 'rational number' as a number which can be expressed as a ratio of two integers which has a terminal decimal number equivolent.
So Pi is
not a rational number, so f(x) != x when x = Pi!!
Basically, rabbit confuses "rational" and "irrational".
Quote from: Rule on February 16, 2006, 09:00 PM
Quote from: rabbit on February 16, 2006, 05:39 PM
But pi is an infinite, non-repeating decimal. I'm using the definition of 'rational number' as a number which can be expressed as a ratio of two integers which has a terminal decimal number equivolent.
So Pi is not a rational number, so f(x) != x when x = Pi!!
Dammit! I never said pi was a rational number! I said it was an infinite non-repeating decimal.
BAH!
I quit.
Quote from: Sorc.Polgara on February 14, 2006, 07:49 PM
f(x) = { x if x is rational
0 if x is irrational }
Quote from: rabbit on February 14, 2006, 09:09 PM
f(x) should be x for integers and non-repeating, infinite decimals (pi, e, etc...), and 0 for everything else.
Shut up! I never said pi was rational, I just ...yeah..shut up!
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roflmao