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

Exercise

Determine the domain of the function:

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

$$-1\leq x< 1$$

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

$$x\neq 1$$

$$-1\leq x< 1$$