Question 1209607
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The domain of the function r(x) = (x^2 + 1)/(1 - x)^2 is (-infty,1) U (1,infty).  What is the range?
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        I will provide another solution, which uses Algebra only

                        and does not use Calculus.



<pre>
Real number "t" belongs to the range of this function if and only if 

    {{{(x^2 + 1)/(1 - x)^2}}} = t for some real x.    (1)


Rewrite (1) in an equivalent form

    {{{(x^2 + 1)/(1 - 2x + x^2)}}} = t,

    {{{x^2 + 1 }}} = {{{t*(1 - 2x + x^2)}}},

     x^2 + 1 = tx^2 - 2tx + t,

     (1-t)x^2 + 2tx + (1-t) = 0.


This last quadratic equation has a real solution IF and ONLY IF the discriminant is non-negative

    d = b^2 - 4ac >= 0,

or

    (2t)^2 - 4*(1-t)*(1-t) >= 0,

    4t^2 - 4*(1-t)^2 >= 0,

    t^2 - 1 + 2t - t^2 >=0,

     2t >= 1,

      t >= 1/2.


At this point, the solution is complete.


<U>ANSWER</U>.  The domain is the set of all real numbers greater than or equal to 1/2.
</pre>

Solved.


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The solution and the answer in the post by Edwin are incorrect.