Precalculus Examples

$f (x) = \frac{1}{\sqrt{| x | - x}}$

Step 1

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

$| x | - x \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Write $| x | - x \geq 0$ as a piecewise.

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Step 2.1.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 2.1.2

In the piece where $x$ is non-negative, remove the absolute value.

$x - x \geq 0$

Step 2.1.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 2.1.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x - x \geq 0$

Step 2.1.5

Write as a piecewise.

${\begin{cases} x - x \geq 0 & x \geq 0 \\ - x - x \geq 0 & x < 0 \end{cases}$

Step 2.1.6

Subtract $x$ from $x$ .

${\begin{cases} 0 \geq 0 & x \geq 0 \\ - x - x \geq 0 & x < 0 \end{cases}$

Step 2.1.7

Subtract $x$ from $- x$ .

${\begin{cases} 0 \geq 0 & x \geq 0 \\ - 2 x \geq 0 & x < 0 \end{cases}$

Step 2.2

Solve $0 \geq 0$ when $x \geq 0$ .

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Step 2.2.1

Since $0 = 0$ , the equation will always be true.

Always true

Step 2.2.2

Find the intersection.

$x \geq 0$

Step 2.3

Solve $- 2 x \geq 0$ when $x < 0$ .

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Step 2.3.1

Divide each term in $- 2 x \geq 0$ by $- 2$ and simplify.

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Step 2.3.1.1

Divide each term in $- 2 x \geq 0$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \leq \frac{0}{- 2}$

Step 2.3.1.2

Simplify the left side.

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Step 2.3.1.2.1

Cancel the common factor of $- 2$ .

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Step 2.3.1.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} \leq \frac{0}{- 2}$

Step 2.3.1.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{0}{- 2}$

Step 2.3.1.3

Simplify the right side.

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Step 2.3.1.3.1

Divide $0$ by $- 2$ .

$x \leq 0$

Step 2.3.2

Find the intersection of $x \leq 0$ and $x < 0$ .

$x < 0$

Step 2.4

Find the union of the solutions.

All real numbers

Step 3

Set the denominator in $\frac{1}{\sqrt{| x | - x}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{| x | - x} = 0$

Step 4

Solve for $x$ .

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Step 4.1

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

${\sqrt{| x | - x}}^{2} = 0^{2}$

Step 4.2

Simplify each side of the equation.

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Step 4.2.1

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

${({(| x | - x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

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

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Step 4.2.2.1.1

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

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Step 4.2.2.1.1.1

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

${(| x | - x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.2.1.1.2.1

Cancel the common factor.

${(| x | - x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2.2

Rewrite the expression.

${(| x | - x)}^{1} = 0^{2}$

Step 4.2.2.1.2

Simplify.

$| x | - x = 0^{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Raising $0$ to any positive power yields $0$ .

$| x | - x = 0$

Step 4.3

Add $x$ to both sides of the equation.

$| x | = x$

Step 4.4

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x = \pm x$

Step 4.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = x$

Step 4.5.2

Move all terms containing $x$ to the left side of the equation.

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Step 4.5.2.1

Subtract $x$ from both sides of the equation.

$x - x = 0$

Step 4.5.2.2

Subtract $x$ from $x$ .

$0 = 0$

Step 4.5.3

Since $0 = 0$ , the equation will always be true.

Always true

Step 4.5.4

Next, use the negative value of the $\pm$ to find the second solution.

$x = - x$

Step 4.5.5

Move all terms containing $x$ to the left side of the equation.

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Step 4.5.5.1

Add $x$ to both sides of the equation.

$x + x = 0$

Step 4.5.5.2

Add $x$ and $x$ .

$2 x = 0$

Step 4.5.6

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 4.5.6.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Step 4.5.6.2

Simplify the left side.

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Step 4.5.6.2.1

Cancel the common factor of $2$ .

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Step 4.5.6.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Step 4.5.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 4.5.6.3

Simplify the right side.

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Step 4.5.6.3.1

Divide $0$ by $2$ .

$x = 0$

Step 4.5.7

The complete solution is the result of both the positive and negative portions of the solution.

$x =, 0$

Step 4.6

Verify each of the solutions by substituting them into $\sqrt{| x | - x} = 0$ and solving.

$x = 0$

Step 5

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

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Step 6