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Domain of One Variable Function – A function with square root – Exercise 5746

Exercise

Determine the domain of the function:

y=(x-2)\sqrt{\frac{1+x}{1-x}}

Final Answer


-1\leq x< 1

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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