Question 958989
<ul><li>For the function to be a true rational function, it must take the form:
{{{n(x)/d(x)}}} where n(x) and d(x) are polynomial functions of x and d(x) is not a constant function.</li><li>For the function to have a natural domain of all real numbers, d(x) must be a function with no (real) zeros.</li><li>For the function to have a range which does not include zero, n(x) must be a function with no (real) zeros.</li><li>For a function to have a range which includes -1, there must be at least one x for which n(x) = -d(x).</li><li>For the remainder of the range, both of the following must be true for all x's excepts the one(s) which make g(x) = -1:<ul><li>|n(x)| < |d(x)|</li><li>the signs of n(x) and d(x) are opposites.</li></ul></li></ul>
Putting all this together I came up with:
{{{g(x) = (-1)/(x^2+1)}}}
This is not the only possible solution. For example, the 1's in the function above can be replaced with any positive number, And I imagine there may be many additional functions which are significantly different from this one and still meet the requirements of the problem.