Precalculus Examples

$f (x) = \sqrt{x - \sqrt{x + 2}}$

Step 1

Set the radicand in $\sqrt{x + 2}$ greater than or equal to $0$ to find where the expression is defined.

$x + 2 \geq 0$

Step 2

Subtract $2$ from both sides of the inequality.

$x \geq - 2$

Step 3

Set the radicand in $\sqrt{x - \sqrt{x + 2}}$ greater than or equal to $0$ to find where the expression is defined.

$x - \sqrt{x + 2} \geq 0$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $x$ from both sides of the inequality.

$- \sqrt{x + 2} \geq - x$

Step 4.2

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

${(- \sqrt{x + 2})}^{2} \geq {(- x)}^{2}$

Step 4.3

Simplify each side of the inequality.

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Step 4.3.1

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

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

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

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

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Step 4.3.2.1.1

Apply the product rule to $- {(x + 2)}^{\frac{1}{2}}$ .

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

Step 4.3.2.1.2

Raise $- 1$ to the power of $2$ .

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

Step 4.3.2.1.3

Multiply ${({(x + 2)}^{\frac{1}{2}})}^{2}$ by $1$ .

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

Step 4.3.2.1.4

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

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Step 4.3.2.1.4.1

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

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

Step 4.3.2.1.4.2

Cancel the common factor of $2$ .

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Step 4.3.2.1.4.2.1

Cancel the common factor.

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

Step 4.3.2.1.4.2.2

Rewrite the expression.

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

Step 4.3.2.1.5

Simplify.

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

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Simplify ${(- x)}^{2}$ .

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Step 4.3.3.1.1

Apply the product rule to $- x$ .

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

Step 4.3.3.1.2

Raise $- 1$ to the power of $2$ .

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

Step 4.3.3.1.3

Multiply $x^{2}$ by $1$ .

$x + 2 \geq x^{2}$

Step 4.4

Solve for $x$ .

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Step 4.4.1

Subtract $x^{2}$ from both sides of the inequality.

$x + 2 - x^{2} \geq 0$

Step 4.4.2

Convert the inequality to an equation.

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

Step 4.4.3

Factor the left side of the equation.

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Step 4.4.3.1

Factor $- 1$ out of $x + 2 - x^{2}$ .

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Step 4.4.3.1.1

Reorder the expression.

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Step 4.4.3.1.1.1

Move $2$ .

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

Step 4.4.3.1.1.2

Reorder $x$ and $- x^{2}$ .

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

Step 4.4.3.1.2

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) + x + 2 = 0$

Step 4.4.3.1.3

Factor $- 1$ out of $x$ .

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

Step 4.4.3.1.4

Rewrite $2$ as $- 1 (- 2)$ .

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

Step 4.4.3.1.5

Factor $- 1$ out of $- (x^{2}) - 1 (- x)$ .

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

Step 4.4.3.1.6

Factor $- 1$ out of $- (x^{2} - x) - 1 (- 2)$ .

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

Step 4.4.3.2

Factor.

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Step 4.4.3.2.1

Factor $x^{2} - x - 2$ using the AC method.

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Step 4.4.3.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 4.4.3.2.1.2

Write the factored form using these integers.

$- ((x - 2) (x + 1)) = 0$

Step 4.4.3.2.2

Remove unnecessary parentheses.

$- (x - 2) (x + 1) = 0$

Step 4.4.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 2 = 0$

$x + 1 = 0$

Step 4.4.5

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 4.4.5.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 4.4.5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 4.4.6

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 4.4.6.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 4.4.6.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 4.4.7

The final solution is all the values that make $- (x - 2) (x + 1) = 0$ true.

$x = 2, - 1$

Step 4.5

Find the domain of $x - \sqrt{x + 2}$ .

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Step 4.5.1

Set the radicand in $\sqrt{x + 2}$ greater than or equal to $0$ to find where the expression is defined.

$x + 2 \geq 0$

Step 4.5.2

Subtract $2$ from both sides of the inequality.

$x \geq - 2$

Step 4.5.3

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

$[- 2, \infty)$

Step 4.6

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