Find the Domain square root of ( square root of x-1)-1

$\sqrt{(\sqrt{x - 1}) - 1}$

Step 1

Set the radicand in $\sqrt{x - 1}$ greater than or equal to $0$ to find where the expression is defined.

$x - 1 \geq 0$

Step 2

Add $1$ to both sides of the inequality.

$x \geq 1$

Step 3

Set the radicand in $\sqrt{(\sqrt{x - 1}) - 1}$ greater than or equal to $0$ to find where the expression is defined.

$(\sqrt{x - 1}) - 1 \geq 0$

Step 4

Solve for $x$ .

Add $1$ to both sides of the inequality.

$\sqrt{x - 1} \geq 1$

To remove the radical on the left side of the inequality, square both sides of the inequality.

${\sqrt{x - 1}}^{2} \geq 1^{2}$

Simplify each side of the inequality.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 1}$ as ${(x - 1)}^{\frac{1}{2}}$ .

${({(x - 1)}^{\frac{1}{2}})}^{2} \geq 1^{2}$

Simplify the left side.

Simplify ${({(x - 1)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x - 1)}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x - 1)}^{\frac{1}{2} \cdot 2} \geq 1^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x - 1)}^{\frac{1}{2} \cdot 2} \geq 1^{2}$

Rewrite the expression.

${(x - 1)}^{1} \geq 1^{2}$

Simplify.

$x - 1 \geq 1^{2}$

Simplify the right side.

One to any power is one.

$x - 1 \geq 1$

Move all terms not containing $x$ to the right side of the inequality.

Add $1$ to both sides of the inequality.

$x \geq 1 + 1$

Add $1$ and $1$ .

$x \geq 2$

Find the domain of $(\sqrt{x - 1}) - 1$ .

Set the radicand in $\sqrt{x - 1}$ greater than or equal to $0$ to find where the expression is defined.

$x - 1 \geq 0$

Add $1$ to both sides of the inequality.

$x \geq 1$

The domain is all values of $x$ that make the expression defined.

$[1, \infty)$

The solution consists of all of the true intervals.

$x \geq 2$

Step 5

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[2, \infty)$

Set-Builder Notation:

${x | x \geq 2}$

Step 6

Find the Domain square root of ( square root of x-1)-1

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