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Thread: When A Function Is to the power of -1

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    When A Function Is to the power of -1

    When A Function Is to the power of -1what does it mean?
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    Quote Originally Posted by abc10 View Post
    When A Function Is to the power of -1what does it mean?
    It can have two meanings.

    If you take the literal meaning for "power of -1", then

    $\displaystyle f^{-1}(x) = \frac{1}{f(x)}$.


    But the more-often used meaning for this notation is the "Inverse Function"

    Basically, if you have a function $\displaystyle f(x)$, then the inverse function $\displaystyle f^{-1}(x)$ is the function whose graph is a reflection along the line $\displaystyle y = x$. So the $\displaystyle x$ and $\displaystyle y$ values have swapped. Also, the domain and range have also swapped.
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    Expanding slightly on what prove it said.

    A function to the power one -1 is the inverse of that function.

    the inverse of a number is a number that multiplied by the original equals 1.
    $\displaystyle x^{-1}x=1$ always.


    -----$\displaystyle \int$
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  4. #4
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    Inverse of a function

    Hello abc10

    Just to expand a little further on the replies so far:

    Yes, $\displaystyle f^{-1}$ denotes the inverse of the function $\displaystyle f$.

    Let's take a simple function like $\displaystyle f(x) = 2x + 5$. Then, for example:
    $\displaystyle f(0) = 5$

    $\displaystyle f(1)=7$

    $\displaystyle f(2)=9$

    ... and so on.
    Then $\displaystyle f^{-1}$ is the function that reverses this process. So:
    $\displaystyle f^{-1}(5) = 0$

    $\displaystyle f^{-1}(7)=1$

    $\displaystyle f^{-1}(9) = 2$

    ... and so on.
    Perhaps you can work out what $\displaystyle f^{-1}$ does? Since $\displaystyle f$ multiplies by $\displaystyle 2$ and then adds $\displaystyle 5$, $\displaystyle f^{-1}$ will do the 'opposite' things in the reverse order. In other words, $\displaystyle f^{-1}$ will:

    • subtract $\displaystyle 5$; and then


    • divide by $\displaystyle 2$.

    So:
    $\displaystyle f^{-1}(x)= \frac{x-5}{2}$
    (Try this function with the values $\displaystyle 5$, $\displaystyle 7$ and $\displaystyle 9$ and check that you get $\displaystyle 0$, $\displaystyle 1$ and $\displaystyle 2$.)

    Grandad
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