Results 1 to 6 of 6

Math Help - Composite Functions (when you are given an implied domain and range)

  1. #1
    Newbie
    Joined
    Mar 2013
    From
    Australia
    Posts
    1

    Composite Functions (when you are given an implied domain and range)

    Hey guys, I am just confused on these types of questions.

    Say if you have a composite function: h(x) = f(g(x)) and you are given an implied domain and an implied range, how do you find out if a number is an element of the domain and range of the outer and inner function?

    For example, if the composite function had an implied domain of [2, 6] and implied range of [-1, infinite) how would you find out if 5 is an element of dom(f) and dom(g) and say 6 is an element of ran(g)?

    What would be the right method to solve this problem?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,712
    Thanks
    1642
    Awards
    1

    Re: Composite Functions (when you are given an implied domain and range)

    Quote Originally Posted by JMoxey View Post
    Say if you have a composite function: h(x) = f(g(x)) and you are given an implied domain and an implied range, how do you find out if a number is an element of the domain and range of the outer and inner function?

    For example, if the composite function had an implied domain of [2, 6] and implied range of [-1, infinite) how would you find out if 5 is an element of dom(f) and dom(g) and say 6 is an element of ran(g)?
    I don't know what "implied domain and an implied range" mean. I have not seen those terms.

    However, in order for f\circ g(a) to exist it is necessary that a\in\text{Dom}(g)~\&~g(a)\in\text{Dom}(f).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,591
    Thanks
    1445

    Re: Composite Functions (when you are given an implied domain and range)

    Quote Originally Posted by Plato View Post
    I don't know what "implied domain and an implied range" mean. I have not seen those terms.

    However, in order for f\circ g(a) to exist it is necessary that a\in\text{Dom}(g)~\&~g(a)\in\text{Dom}(f).
    And also that the range of g(x) is completely contained in the domain of f(x).
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,712
    Thanks
    1642
    Awards
    1

    Re: Composite Functions (when you are given an implied domain and range)

    Quote Originally Posted by Prove It View Post
    And also that the range of g(x) is completely contained in the domain of f(x).
    That may not be the case.
    Consider f(x)=\sqrt{x}~\&~g(x)=x^3.
    We can have the function h(x)=f\circ g(x)=\sqrt{x^3} with \text{Dom}(h)=[0,\infty)=\text{Dom}(f)\cap\text{Rng}(g) but \text{Rng}(g)\not\subset\text{Dom}(f).
    Last edited by Plato; March 23rd 2013 at 02:40 PM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,591
    Thanks
    1445

    Re: Composite Functions (when you are given an implied domain and range)

    Ah, no we can't, at least not if we're assuming \displaystyle f: \mathbf{R} \to \mathbf{R}, f(x) = \sqrt{x} and \displaystyle g: \mathbf{R} \to \mathbf{R}, g(x) = \sqrt{x}.

    \displaystyle \sqrt{x^3} is not defined for any \displaystyle x < 0, so that means \displaystyle f \circ g(x) does not exist for any \displaystyle g(x) which are negative. In this case for f \circ g(x) to exist, we require restricting the domain of \displaystyle g(x) so that the range will be \displaystyle [0, \infty).
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,712
    Thanks
    1642
    Awards
    1

    Re: Composite Functions (when you are given an implied domain and range)

    Quote Originally Posted by Prove It View Post
    Ah, no we can't, at least not if we're assuming \displaystyle f: \mathbf{R} \to \mathbf{R}, f(x) = \sqrt{x} and \displaystyle g: \mathbf{R} \to \mathbf{R}, g(x) = \sqrt{x}.
    \displaystyle \sqrt{x^3} is not defined for any \displaystyle x < 0, so that means \displaystyle f \circ g(x) does not exist for any \displaystyle g(x) which are negative. In this case for f \circ g(x) to exist, we require restricting the domain of \displaystyle g(x) so that the range will be \displaystyle [0, \infty).
    No one ever said anything about f:R\to R. The example simply said f(x)=\sqrt{x} thus its domain is understood to be [0,\infty). We can find this example in almost any Pre-Caculus textbook. It is a standard question to ask about the domain of a composition of two functions.

    The domain of f\circ g is \text{Dom}(f)\cap\text{Rng}(g).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Range of composite functions
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: September 11th 2011, 11:04 AM
  2. Replies: 2
    Last Post: February 19th 2010, 05:50 AM
  3. Range of Composite Functions
    Posted in the Pre-Calculus Forum
    Replies: 5
    Last Post: June 17th 2009, 08:02 AM
  4. Proofs: Domain of composite functions
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: April 20th 2009, 02:49 PM
  5. Please help with domain of composite functions!
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: September 17th 2008, 01:35 PM

Search Tags


/mathhelpforum @mathhelpforum