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Homework Help: Rational functions and link with direct substitution property

  1. Oct 19, 2013 #1
    1. The problem statement, all variables and given/known data

    I know the direct substitution property in calculus. But, the definition of a rational function still confuses me.

    For example, are these rational functions:



    The denominator of the first one could cancel. So, is there still a ratio? Is it a rational function?

    The second one has a root in the numerator. The exponents of are not integers. Is it still a rational function?

    Please help me. I do not understand the definition.
  2. jcsd
  3. Oct 19, 2013 #2


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    Rational expression: ratio of two polynomials; since 1 is a polynomial of order zero, y=x+1 is also a rational expression, as is y=1/x ... as long as you exclude zeros in the denominator.

    A rational function is just a function which can be written as a rational expression. Your second example is not a rational expression, so y is not a rational function.
  4. Oct 19, 2013 #3
    Thank you for the fast reply.
    But, what do you mean by rational expression? I do not quite get it.

    What about if there was an absolute value in the function?

    EDIT: I understand why the second one is not a rational expression! Thank you. I still have a doubt about absolute values.
    Last edited: Oct 19, 2013
  5. Oct 19, 2013 #4


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    Rational expression: from ratio of expressions - in this case the individual expressions are always polynomials, so a rational expression is simply the ratio of two polynomials.

    Your text may extend this to include absolute value of a ratio of polynomials, but that is not what is usually meant. But you always need to look at (and understand, and remember) the definition which is provided.
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