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Math Help - Range

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    MHF Contributor Mathstud28's Avatar
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    Range

    Does anyone know a good way of finding the range of a function...I usually either man it out and look for the wholes and check its horizontal endpoints...or if its a simple function I find its inverse and check that functions domain
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    Quote Originally Posted by Mathstud28 View Post
    Does anyone know a good way of finding the range of a function...I usually either man it out and look for the wholes and check its horizontal endpoints...or if its a simple function I find its inverse and check that functions domain
    One word: Graph it.
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    Quote Originally Posted by Mathstud28 View Post
    Does anyone know a good way of finding the range of a function...I usually either man it out and look for the wholes and check its horizontal endpoints...or if its a simple function I find its inverse and check that functions domain
    Here is another way. The range of a function are all y so that the equation f(x) = y has a solution in the domain.
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    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    Here is another way. The range of a function are all y so that the equation f(x) = y has a solution in the domain.
    Yeah that is how I check the possibility for a function to not have an element of the range...but it dose not help me for finding the range...for example I could use that method to check whether or not 1 is in the range by saying 1=f(x) and seeing if there is an x value that satisfies that equation...but it doesnt help me to go o well since f(x)=randomfunction then the range is this
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    Quote Originally Posted by Mathstud28 View Post
    Yeah that is how I check the possibility for a function to not have an element of the range...but it dose not help me for finding the range...for example I could use that method to check whether or not 1 is in the range by saying 1=f(x) and seeing if there is an x value that satisfies that equation...but it doesnt help me to go o well since f(x)=randomfunction then the range is this
    Here is an example. Let y=2e^{-x^2}, the domain is of course all numbers. The range are all y so that we can solve the equation y=2e^{-x^2} for x in the domain. We begin by writing, (y/2) = e^{-x^2} but that can only have a solution if (y/2) > 0 (because the exponentinal is positive). Which means if y/2>0 then \ln (y/2) = -x^2 \implies - \ln (y/2) = x^2. Now to solve this equation we require - \ln (y/2) \geq 0 \implies \ln (y/2) < 0 \implies 0<y/2\leq 1\implies 0 < y\leq 2. For those values we can solve the equation.

    Try this example. Find the range of y = x/(x^2+1).
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