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Thread: Range of function

  1. #1
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    Range of function

    Hi

    What will be the range of the function f(x) = 1-1/x .

    1. The domain is given as [0, infinity)

    2. The domain is given as [0, infinity]

    I am primarily confused if "1" will be included or not? The limit of the function tends to 1 but it will never actually reach 1. So, will the range be [-infinity,1) or [-infinity,1] or (-infinity,1) or (-infinity,1]

    thanks
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  2. #2
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    Quote Originally Posted by champrock View Post
    Hi

    What will be the range of the function f(x) = 1-1/x .

    1. The domain is given as [0, infinity)

    2. The domain is given as [0, infinity]

    I am primarily confused if "1" will be included or not? The limit of the function tends to 1 but it will never actually reach 1. So, will the range be [-infinity,1) or [-infinity,1] or (-infinity,1) or (-infinity,1]

    thanks
    Hint: Find the range of the function $\displaystyle f(x) = \frac{1}{x}$ in the domain $\displaystyle x \geq 0$.

    The multiplication by $\displaystyle -1$ gives a reflection in the x-axis, and the addition of 1 is a translation of 1 unit vertically upward.
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  3. #3
    MHF Contributor Amer's Avatar
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    I write the answer but when I see the member above me give you a hint I edit it

    note you can't write the interval closed from the direction of infinity
    like this

    (0,infinity]

    the correct like this

    (0,infinity)
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  4. #4
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    Quote Originally Posted by Amer View Post
    I write the answer but when I see the member above me give you a hint I edit it

    note you can't write the interval closed from the direction of infinity
    like this

    (0,infinity]

    the correct like this

    (0,infinity)
    hmm ok . (0,infinity) is because infinity cant be included in the set right? Its not any proper number.

    @Prove It : I found the range. Its coming out to be -infinity to 1. But I am not sure if 1 is actually included or not because
    Lim 1/x (x tends to infinity) ---> 0
    so 1-1/x = 1-0 = 1
    but
    But 1/(some very large number ) ----> some very tiny number like 0.000000000045
    So, 1-1/x = 1-some very tiny number. = 0.99999999999999434

    SO, the range is (-infinity,1) or (-infinity,1] ?
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  5. #5
    MHF Contributor Amer's Avatar
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    the first one (-infinity,1)

    thanks to prove it
    Last edited by Amer; Jun 6th 2009 at 04:43 AM.
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  6. #6
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    Quote Originally Posted by champrock View Post
    hmm ok . (0,infinity) is because infinity cant be included in the set right? Its not any proper number.

    @Prove It : I found the range. Its coming out to be -infinity to 1. But I am not sure if 1 is actually included or not because
    Lim 1/x (x tends to infinity) ---> 0
    so 1-1/x = 1-0 = 1
    but
    But 1/(some very large number ) ----> some very tiny number like 0.000000000045
    So, 1-1/x = 1-some very tiny number. = 0.99999999999999434

    SO, the range is (-infinity,1) or (-infinity,1] ?
    Let's put it this way...

    In the domain $\displaystyle x > 0$, (since $\displaystyle x \neq 0$ as pointed out earlier), the range of the function $\displaystyle \frac{1}{x}$ is $\displaystyle y > 0$, or in interval notation, $\displaystyle (0, \infty)$.

    Notice that as $\displaystyle x \to \infty, y \to 0$. BUT since we can never reach $\displaystyle \infty$, $\displaystyle y$ will never reach $\displaystyle 0$.

    Now the reflection in the $\displaystyle x$ axis means that the range of $\displaystyle -\frac{1}{x}$ becomes $\displaystyle y < 0$, or to use interval notation, $\displaystyle (-\infty, 0)$.

    The translation of 1 unit vertically upward means that the range of $\displaystyle 1 - \frac{1}{x}$ is $\displaystyle y < 1$, or $\displaystyle (-\infty, 1)$ in interval notation.


    So to answer your question, you do NOT include the 1 in the range.
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