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Math Help - f functions and domain/range

  1. #1
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    f functions and domain/range

    hi

    let f(x) = x/(x-1). determine f(f(x)) simplify answer, and find domain/range of f(f(x))

    is the question i got.

    help please!

    also not sure if posted in the right forum?
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  2. #2
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    Re: f functions and domain/range

    Quote Originally Posted by tankertert View Post
    hi

    let f(x) = x/(x-1). determine f(f(x)) simplify answer, and find domain/range of f(f(x))

    is the question i got.

    help please!

    also not sure if posted in the right forum?
    f(f(x))=\frac{\frac{x}{x-1}}{\frac{x}{x-1}-1}=\frac{\frac{x}{x-1}}{\frac{1}{x-1}}=x

    \text{domain } : x \in (-\infty,+\infty)

    \text{range } : f(f(x)) \in (-\infty,+\infty)
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  3. #3
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    Re: f functions and domain/range

    This might not be exactly true.

    The domain of  f(x) = \frac{x}{x-1} is  \mathbb{R} \setminus \{1\}
    and its range is really  \mathbb{R} (infinity shall be included in the real numbers).

    But then again, f(f(x)) has the above mentioned domain (since the image of  f: \mathbb{R} \rightarrow \mathbb{R} is  \mathbb{R} ),
    and then you map the real numbers again, in the same way, to itself. And again, the value is not defined at x = 1 (you can really not define it there, because
    the left and right limits are different).

    So I'd say it's a little tricky here and would nevertheless say that the domain is  \mathbb{R} \setminus \{1\} .

    Of course you have as effective function the identity mapping, however since you cannot divide by zero, the simplification holds ONLY IF x is NOT 1.

    That's the point.
    I may be corrected if I'm wrong, however I think that this is the correct answer.
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  4. #4
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    Re: f functions and domain/range

    Yes, mastermind is correct. f(f(x))= (x/(x-1))/(1/(x-1)) for all x for which it is defined- all x except 1. That reduces to f(x)= x for those values of x- x not equal to 1.

    A simpler example of that is (x- 1)/(x^2- 1)= (x- 1)/((x- 1)(x+ 1))= 1/(x+ 1) for all x except 1. The left side is not defined for x= 1 but the right side is so they are not equal for x= 1.
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  5. #5
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    Re: f functions and domain/range

    graph of f[f(x)] ... note the discontinuity at x = 1
    Attached Thumbnails Attached Thumbnails f functions and domain/range-f-f-x-.jpg  
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