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Home/ Questions/Q 1504
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Janaka Rodrigo
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Janaka RodrigoLevel 3
Asked: February 23, 20232023-02-23T23:13:23+05:30 2023-02-23T23:13:23+05:30In: Algebra ( වීජ ගණිතය )

Composite functions

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1)What are the conditions required to be satisfied to exist the composite function f(g(x))?

2)Do we need range of g to be a subset of domain of f or is intersection of mentioned range and domain sufficient and what is the domain of f(g(x)) ?

3)For what real values of k can f(g(x)) be defined?

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  1. Janaka Rodrigo Level 3 English medium advanced level mathematics teacher at St Benedicts College Colombo 13
    2023-02-27T15:05:48+05:30Added an answer on February 27, 2023 at 3:05 pm

    1)When f(x) and g(x) are given as expressions you can just replace x in f by g(x) and it gives an expression for f(g(x)).

    Let f(x) = x²  , g(x) =  √(x – 1)                                      f(g(x)) = x – 1 and f(g(0))= -1

    2)When domains of f and g given you can find the range of g , and f(g(x)) exists if the range of g is a subset of domain of f  . In this case domain of f(g(x)) is same as the domain of the function g.

    In the previous example if domain of f given as all real numbers and domain of g as real numbers not less than 1 ,  range of g is set of positive real numbers including 0 and it is subset of domain of f . Therefore f(g(x)) can be defined. But f(g(0)) can not be defined because 0 is not in the domain of the composite function. Since range of f is not a subset of domain of g ,  g(f(x)) can not be defined.

    3)If you are asked to find f(g(x)) with it’s domain without giving domains of f and g , you need to restrict the domain of g in order to make range of g to be a subset of domain of f.

    Let f(x) = √(x -1) , g(x) = x²                                       here you need to restrict domain of g to             (-∞ , 1] ∪  [ 1 , ∞ ) in order have range of g that is [ 1  ,  ∞ ) as a subset of domain of f.

    Hope this answer would be sufficient to get the clear picture of the issue. I think better way to get this fully understood is to practice for few more problems.  Appreciate your suggestions regarding this answer. Because of a system error sometimes your reply can be taken as another answer.

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  2. Gaya Nayomi Level 1 ( Top Rated ) Senior Lecturer-University of Sri Jayewardenepura
    2023-03-18T20:17:40+05:30Added an answer on March 18, 2023 at 8:17 pm
    This answer was edited.
    1. The answer you have provided to your own question #1 is not the answer to the question you have asked. You have explained a procedural step to find the composite function f(g(x)) (when it exists) when f(x) and g(x) are given. But your question (which is not clearly stated), we think, is about the conditions under which a composite function exists. Assuming it is “What are the conditions required for f(g(x)) to exist?”, the answer is as follows:

              Conditions for f(g(x)) to exist are,

              * the range of g(x) must be a subset of the domain of f(x)

                    Or

               *f(g(x)) must be defined with the domain of g(x) restricted so that its range is a subset of the domain of f(x).

         2. This has been already answerd in the response to #1.

         3. This question makes no sense. (what’s k?)

    ~MetaHub Panel for the Teachers’ Forum

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    • Janaka
      2023-03-18T22:56:15+05:30Replied to answer on March 18, 2023 at 10:56 pm

      Thank you very much for helping to clear doubts. In 1 st case I have not mentioned functions , therefore don’t we have to treat them as just polynomial expressions? In the 3 rd case I can not see k there , could you please explain little bit more Madam?

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      • Janaka
        2023-03-18T23:17:12+05:30Replied to answer on March 18, 2023 at 11:17 pm

        Really sorry for not pointing it out, that should be x instead of k Madam. Hope you may excuse me,  still I couldn’t find how to use options like editing , deleting in my android.

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