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!

F at g(a) versus f o g at a?

  1. Oct 9, 2013 #1
    1. The problem statement, all variables and given/known data

    I have to understand this to understand my homework. What is the difference between these two function compositions?

    2. Relevant equations

    f at g(a)

    f o g at a

    3. The attempt at a solution

    Let's make f(a) = a^2 and g(a) = 3a.
    It seems that "f at g(a)" means f(g(a)) which equals (3a)^2.
    And it also seems like "f o g at a" means f(g(a)) which equals (3a)^2.
     
  2. jcsd
  3. Oct 9, 2013 #2

    Mark44

    Staff: Mentor

    They're both the same. It's just different notation for the same thing.
     
  4. Oct 9, 2013 #3
    Are you sure? Here is the reason why I asked this. In my notes, it says, "Corollary 3.1.7 (Composition of Continuous Functions): Suppose g is continuous at a and f is continuous at g(a). Then f ◦g is continuous at a".

    Maybe this whole corollary somehow changes the meaning of the two function compositions?
     
  5. Oct 9, 2013 #4

    Mark44

    Staff: Mentor

    f o g is a function in its own right. It's defined as: (f o g)(x) = f(g(x)).
     
  6. Oct 9, 2013 #5

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    No, it doesn't change anything. The domain of ##h = f\circ g## is the domain of ##g##, so it makes sense to think about continuity of ##h## at points ##a## in the domain of ##g## where ##g## is continuous. I'm not really sure what is bothering you about this.
     
  7. Oct 9, 2013 #6
    It bothers me because then the corollary is essentially saying that if g is continuous at x and f(g(x)) is continuous at x, then f(g(x)) is also continuous at x. There would be no reason to have the last part.
     
  8. Oct 9, 2013 #7

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Let me give an example where g is not continuous to show what they mean. Suppose f(x) = x2 and g(x) = 1 for all x except g(0) = 4. Then f is continuous at 4 so f is continuous at g(0), but f(g(x)) is not continuous at 0 (because g is not continuous at 0).

    Hopefully this clears things up
     
  9. Oct 9, 2013 #8

    Mark44

    Staff: Mentor

    That's not what it's saying. It's saying that if g is continuous at a, and f is continuous at g(a), then f o g is continuous at a. That's different from what you wrote.
     
  10. Oct 9, 2013 #9
    Would you agree that "f is continuous at g(a)" means the exact same thing as "f ◦g is continuous at a"? If so, then we can just change the former with the latter and then the corollary reads,
    "Suppose g is continuous at a and f ◦g is continuous at a. Then f ◦g is continuous at a".

    Normally I would question the notes, but my professor said that he made them over 20 years ago with another mathematician, and they have been using them ever since.
     
  11. Oct 9, 2013 #10

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    No, I don't agree it's the same thing. f is continuous at g(a) is talking about the continuity of the function f, at a value in the domain of f, which happens to be g(a). f◦g being continuous at a is talking about the continuity of f◦g at a value in its domain. They're referring to the continuity of two different functions at points in two different domains, they do not mean the same thing. Read my post again I give an example of where f IS continuous at g(0), but f◦g is NOT continuous at 0, so those two concepts must be different.

    The value of f at g(a) is the same as the value of f◦g at a, but the value of f NEAR g(a) is not the same thing as the value of f◦g near a, and continuity is talking about the latter, not the former.
     
  12. Oct 9, 2013 #11
    I know; I changed "f is continuous at g(x)" and "f ◦g is continuous at a" to "f(g(x))" since I am told that they are all equivalent, and then the corollary would exactly mean, "If g is continuous at x and f(g(x)) is continuous at x, then f(g(x)) is also continuous at x.
     
  13. Oct 9, 2013 #12
    Ohhhh, that's what was screwing me up. Now I see the point, thanks!
     
  14. Oct 9, 2013 #13
    Thank-you everyone :)
     
  15. Oct 9, 2013 #14

    Mark44

    Staff: Mentor

    No, they are not all equivalent, and your change has a different meaning from what you started with. The statement in your notes is exclusively about functions: namely, f, g, and f o g.

    g(x) is not a function - it's the value of the function at a number x in the domain of f. Likewise, f(g(x)) is also not a function - it's the value of the function f o g at a number in its domain.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted