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

Show 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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