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    How do you find the range of fractional functions e.g f(x) = 5x/x^1+1 ?


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    Do you mean this: f(x)=\frac{5x}{x^{2}+1}?

    You could try finding the minimum and maximum of that function (when you differentiate and set =0).
    You could also try trial-and-error and put numbers in your calculator to see what is happening, and try and make little sketch.
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    this example is where the denominator is defined for all x - so you know there`s no problem with vertical asymptotes.

    For this particular example, y is at it`s maximum and minimum when x is at its lowest possible minimum : x^{2}=1

    For most other rational functions, you`d have to examine the behaviour of y as x approached +/- infinity and the vertical asymptotes.
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    Put the function as y=f(x), take the denominator on the left hand side to basically form a quadratic in terms of x. Put the condition that all roots of the quadratic must be real to obtain a range for y(always works )
 
 
 
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