1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Intro To real analysis problem

  1. Apr 23, 2012 #1
    1. The problem statement, all variables and given/known data
    a) Find f ([0,3]) for the following function:
    f(x)=1/3 x^3 − x + 1

    b) Consider the following function :
    f(x) = e^(−ax) (e raised to the power of '-a' times 'x') a, x ∈ [0,∞)
    Find values of a for which f is a contraction .

    c) Prove that for all x,y ≤ 0 | 2^x −2^y | ≤ |x−y|
    Last edited: Apr 23, 2012
  2. jcsd
  3. Apr 23, 2012 #2


    User Avatar
    Science Advisor

    Confuse about what? Do you know the definition of "f(A)" for f a function and A a set? It is: { f(x)| x in set A}. You might find it simplest to graph the function. The draw vertical lines at x= 0 and x= 3. Where those vertical lines cross the graph, draw horizontal lines to make a rectangle. f(A) is the set of all y values inside that rectangle.
  4. Apr 23, 2012 #3
    I think I have done part A properly. When you are taking f of a set, you are simply mapping each value in the set to another set right?
    for a, I got the set {1, 1/3, 5/3, 7}
    is this correct?

    For part B I am confused because, well to be honest im terrible at proofs ( you can imagine how this class has been going for me ). I know what a contraction is, I simply just do not know how to approach part B.

    For part C, I was able to get a little attempt going, but I seem to have gone astray. Once again the problem is Approaching the proof. My mind becomes all jumbled trying to approach this stuff.
  5. Apr 23, 2012 #4


    User Avatar
    Science Advisor

    No, isn't! Those {f(0), f(1), f(2), f(3)}. That would be correct if were {0, 1, 2, 3}. It is not. "[0, 3]" means "the set of all real numbers from 0 to 3, inclusive".

    Part B does not ask for a proof! I'm glad you know what a contraction is. What is the precise definition of "contraction"? Typically in both problems and proofs, you can use the exact words of a degfinition. Specifically, to show that something is a "contraction" you show that it satisfies the definition.

    In (C) did you notice the comdition that x and y are both negative? What can you say about 2x and 2y when x and y are negative?
  6. Apr 23, 2012 #5
    Thank you for your reply!

    Well for part a, I did something similar in my notes ( I think ). Unless I took notes wrong, the professor took the min and max of the interval ( so 0 and 3 respectively ), solved f at those points, then took f prime, set it equal to zero, then solved for x?
    Is this a step in the right direction? Could be very wrong.

    Sorry you are right for part B, I do not know what I was thinking. Ok, so I know a contraction is defined as
    | f(x) - f(y) | <= n | x - y | For some 0 < n < 1
    So would I set
    f(x) = e^(-a_1x)
    f(y) = e^(-a_2x)
    Then use mean value theorum? If so how would I apply it. I seem to not be able to get past the definition.

    For part c I did infact recognize that x and y are negative. This means that 2^x and 2^y will be 0 < 2^x, 2^y <=1
    This also means that -1 < 2^x - 2^y < 1
    => 0 <= | 2^x - 2^y | < 1
    also | x - y | >= 0 <= | 2^x - 2^y |... which yeilds | x - y | >= | 2^x - 2^y |
    oh.... awkward.... :)
    figured it out while typing it in the forum :) I'll keep the results though. they are right? right?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook