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I am asked to argue whether or not the following two functions are well-defined (textbook definition: a) define $y$ for all $x$ in domain, and b) any is mapped to exactly one y). Both of the below functions are functions from $\mathbb{Q}$ to $\mathbb{Q}$.

$$f\left(\frac{p}{q}\right) = \frac{p+1}{q}$$

$$g\left(\frac{p}{q}\right) = \frac{p+q}{p-q}$$

My argument is that since $0$ is a rational number, we can take, for $f$, $p=0$ and $q=x$ and the function will not be defined. Similarly, we can take $p=q=0$ for $g$, and the function, again, will not be defined.

But the argument seems to be too easy. Is there something I am missing that won't allow me to use these two counter examples?

Thanks!

  • 0
    For the first one you probably means $p=0$ and $q=x$, otherwise you divide by $0$. For the second one, if $p=0$ and $q=0$, what does $\frac{p}{q}=\frac{0}{0}$ mean?2012-06-03
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    You are correct, typo.2012-06-03
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    As an addendum, the notion of a "homogeneous polynomial" is important to defining things this way. A homogeneous polynomial is one where the total degree of every monomial is the same, such as $x^2 + 2 xy + 3 y^2$. When you're defining $f(x/y)$ as a rational function in $x$ and $y$, the numerator and denominator must be homogeneous polynomials of the same degree.2012-06-03

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