Exercise
Determine the domain of the function:
y=(x-2)\sqrt{\frac{1+x}{1-x}}
Final Answer
Solution
Let’s find the domain of the function:
y=(x-2)\sqrt{\frac{1+x}{1-x}}
Because there is a denominator, the denominator must be different from zero:
1-x\neq 0
x\neq 1
Also, there is a square root, so we need the expression inside the root to be non-negative:
\frac{1+x}{1-x}\geq 0
This inequality is equivalent to the inequality:
(1+x)(1-x)\geq 0
Therefore, we solve the latter:
(1+x)(1-x)\geq 0
Open brackets:
1-x^2\geq 0
It is a square inequality. The roots of the quadratic equation:
1-x^2=0
are
x=\pm 1
Because we are looking for the section above the x-axis or on it and the parabola “cries”, we get that the solution of the inequality is
-1\leq x\leq 1
Finally, let us not forget that we also demanded that
x\neq 1
So together the answer is
-1\leq x< 1
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