Math- Rational Functions

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1. Some rational functions have asymptotes, others have holes, and some have both. Explain how you can identify, without graphing, which graphical features a rational function will have.

Can someone explain this thoroughly ? I don't understand. thanks in advance.

  • Math- Rational Functions -

    the culprit for asymptotes and holes is the denominator.

    If for some value of the variable, then denominator is zero, but the numerator is NOT zero, you will have an asymptote.

    If for some value of the variable, then denominator is zero, but the numerator is ALSO zero, you will have a hole

    e.g. y = (x-2)/(x^2 - 4)

    notice this reduces to y = 1/(x+2)

    so if x = 2 in the original we get 0/0, so there is a hole at (2,1/4)
    if x = -2 we get -4/0 in the original, so x = -2 is an asymptote

  • Math- Rational Functions -

    how would we know if it has both holes and asymptotes?

  • Math- Rational Functions -

    What is the domain and range of 2x^2-18/x^2+3x-10

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