Calculus Examples

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

Step 1

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

$x \geq 0$

Step 2

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

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

Step 3

Solve for $x$ .

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Step 3.1

Subtract $2$ from both sides of the inequality.

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

Step 3.2

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

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

Step 3.3

Simplify each side of the inequality.

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Step 3.3.1

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

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

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

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Step 3.3.2.1.1

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

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

Step 3.3.2.1.2

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

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

Step 3.3.2.1.3

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

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

Step 3.3.2.1.4

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

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Step 3.3.2.1.4.1

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

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

Step 3.3.2.1.4.2

Cancel the common factor of $2$ .

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Step 3.3.2.1.4.2.1

Cancel the common factor.

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

Step 3.3.2.1.4.2.2

Rewrite the expression.

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

Step 3.3.2.1.5

Simplify.

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

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

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

$x \geq 4$

Step 3.4

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

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Step 3.4.1

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

$x \geq 0$

Step 3.4.2

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

$[0, \infty)$

Step 3.5

Use each root to create test intervals.

$x < 0$

$0 < x < 4$

$x > 4$

Step 3.6

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 3.6.1

Test a value on the interval $x < 0$ to see if it makes the inequality true.

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Step 3.6.1.1

Choose a value on the interval $x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 3.6.1.2

Replace $x$ with $- 2$ in the original inequality.

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

Step 3.6.1.3

The left side is not equal to the right side, which means that the given statement is false.

False

Step 3.6.2

Test a value on the interval $0 < x < 4$ to see if it makes the inequality true.

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Step 3.6.2.1

Choose a value on the interval $0 < x < 4$ and see if this value makes the original inequality true.

$x = 2$

Step 3.6.2.2

Replace $x$ with $2$ in the original inequality.

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

Step 3.6.2.3

The left side $0.58578643$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 3.6.3

Test a value on the interval $x > 4$ to see if it makes the inequality true.

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Step 3.6.3.1

Choose a value on the interval $x > 4$ and see if this value makes the original inequality true.

$x = 6$

Step 3.6.3.2

Replace $x$ with $6$ in the original inequality.

$2 - \sqrt{6} \geq 0$

Step 3.6.3.3

The left side $- 0.44948974$ is less than the right side $0$ , which means that the given statement is false.

False

Step 3.6.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < 0$ False

$0 < x < 4$ True

$x > 4$ False

$x < 0$ False

$0 < x < 4$ True

$x > 4$ False

Step 3.7

The solution consists of all of the true intervals.

$0 \leq x \leq 4$

Step 4

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

Interval Notation:

$[0, 4]$

Set-Builder Notation:

${x | 0 \leq x \leq 4}$

Step 5