.
In a normal mathematical language, this post must sound in this way:
Find the domain of the function fourth root of (1-x)/(2-x).
Solution
f(x) = . (1)
The domain is the set of those real numbers, where the expression under the root is non-negative:
>= 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 is non-negative in this domain
and the set x <= 1 DOES BELONG 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 is negative in this domain
and the set 1 < x < 2 DOES NOT BELONG 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 is positive in this domain
and the set x > 1 DOES BELONG to the set of solutions of inequality (2).
Answer. The domain of the given function is the set (,1] U (2,).
Solved.
----------------
To see more examples of solved problems on inequalities and finding function domains, look into these relevant lessons
- Solving problems on quadratic inequalities,
- Solving inequalities for high degree polynomials factored into a product of linear binomials
- Solving inequalities for rational functions with numerator and denominator factored into a product of linear binomials
- Solving inequalities for rational functions with non-zero right side
- Advanced lesson on inequalities
- Domain of a function which involves a quadratic polynomial under the square root operator,
- Domain of a function containing high degree polynomial under the square root operator
- Domain of a function which is the square root of a rational function
in this site.
Also, you have this free of charge online textbook in ALGEBRA-I in this site
- ALGEBRA-I - YOUR ONLINE TEXTBOOK.
The referred lesson is the part of this online textbook under the topic "Inequalities".
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.