Question 1100913
.
In a normal mathematical language, this post must sound in this way:

<pre>   
    Find the domain of the function  fourth root of (1-x)/(2-x).
</pre>

<U>Solution</U>


<pre>
f(x) = {{{root(4,(1-x)/(2-x))}}}.      (1)

The domain is the set of those real numbers, where the expression under the root is non-negative:

    {{{(1-x)/(2-x)}}} >= 0.      (2)

So, we need to find the solution set of this inequality.


The rational function (2) has two critical points: x= 1  and  x= 2  (in ascending order).

1)  If x <= 1, then both binomial are non-positive:  

              1-x <= 0  and  2-x < 0.

              Hence, the ratio  {{{(1-x)/(2-x)}}} is non-negative in this domain 

              and the set  x <= 1  <U>DOES BELONG</U>  to the set of solutions to inequality (2).



2)  If 1 < x <2, then the binomial 1-x is positive  and  the binomial 2-x is negative:

              1-x > 0   and  2-x < 0.

              Hence, the ratio  {{{(1-x)/(2-x)}}} is negative in this domain 

              and the set  1 < x < 2  <U>DOES NOT BELONG</U>  to the set of solutions to equation (2).



2)  If x > 2, then both binomials  1-x  and  2-x  are greater than 0:  

              1-x > 0  and  2-x > 0.

              Hence, the ratio  {{{(1-x)/(2-x)}}} is positive in this domain 

              and the set  x > 1  <U>DOES BELONG</U>  to the set of solutions of inequality (2).
</pre>

<U>Answer</U>.  The domain of the given function is the set  ({{{-infinity}}},1] U (2,{{{infinity}}}).



Solved.



----------------
To see more examples of solved problems on inequalities and finding function domains, look into these relevant lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Inequalities/Solving-problems-on-quadratic-inequalities.lesson>Solving problems on quadratic inequalities</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Inequalities/Solving-inequalities-for-polynomials-factored-to-a-product-of-linear-binomials.lesson>Solving inequalities for high degree polynomials factored into a product of linear binomials</A>  

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Inequalities/Solving-inequalities-for-rat-functions-with-num-and-denom-factored-into-a-product-of-linear-binomials.lesson>Solving inequalities for rational functions with numerator and denominator factored into a product of linear binomials</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Inequalities/Solving-inequalities-for-rational-functions-with-non-zero-right-side.lesson>Solving inequalities for rational functions with non-zero right side</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Inequalities/Advanced-lesson-on-inequalities.lesson>Advanced lesson on inequalities</A>


&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Functions/Finding-domain-of-a-function.lesson>Domain of a function which involves a quadratic polynomial under the square root operator</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Functions/Domain-of-a-function-containing-high-degree-polynomial-under-the-square-root.lesson>Domain of a function containing high degree polynomial under the square root operator</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Functions/Domain-of-a-function-which-is-the-square-root-of-a-rat-function.lesson>Domain of a function which is the square root of a rational function</A>

in this site.



Also, &nbsp;you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lesson is the part of this online textbook under the topic &nbsp;"<U>Inequalities</U>".



Save the link to this online textbook together with its description


Free of charge online textbook in ALGEBRA-I

https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson


to your archive and use it when it is needed.