Calculus Examples

$f (x) = \frac{1}{\sqrt[4]{x^{2} - 5 x}}$

Step 1

Set the radicand in $\sqrt[4]{x^{2} - 5 x}$ greater than or equal to $0$ to find where the expression is defined.

$x^{2} - 5 x \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Convert the inequality to an equation.

$x^{2} - 5 x = 0$

Step 2.2

Factor $x$ out of $x^{2} - 5 x$ .

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Step 2.2.1

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

$x \cdot x - 5 x = 0$

Step 2.2.2

Factor $x$ out of $- 5 x$ .

$x \cdot x + x \cdot - 5 = 0$

Step 2.2.3

Factor $x$ out of $x \cdot x + x \cdot - 5$ .

$x (x - 5) = 0$

Step 2.3

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

$x = 0$

$x - 5 = 0$

Step 2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.5

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 2.5.2

Add $5$ to both sides of the equation.

$x = 5$

Step 2.6

The final solution is all the values that make $x (x - 5) = 0$ true.

$x = 0, 5$

Step 2.7

Use each root to create test intervals.

$x < 0$

$0 < x < 5$

$x > 5$

Step 2.8

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 2.8.1

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

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Step 2.8.1.1

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

$x = - 2$

Step 2.8.1.2

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

${(- 2)}^{2} - 5 \cdot - 2 \geq 0$

Step 2.8.1.3

The left side $14$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 2.8.2

Test a value on the interval $0 < x < 5$ to see if it makes the inequality true.

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Step 2.8.2.1

Choose a value on the interval $0 < x < 5$ and see if this value makes the original inequality true.

$x = 2$

Step 2.8.2.2

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

${(2)}^{2} - 5 \cdot 2 \geq 0$

Step 2.8.2.3

The left side $- 6$ is less than the right side $0$ , which means that the given statement is false.

False

Step 2.8.3

Test a value on the interval $x > 5$ to see if it makes the inequality true.

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Step 2.8.3.1

Choose a value on the interval $x > 5$ and see if this value makes the original inequality true.

$x = 8$

Step 2.8.3.2

Replace $x$ with $8$ in the original inequality.

${(8)}^{2} - 5 \cdot 8 \geq 0$

Step 2.8.3.3

The left side $24$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 2.8.4

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

$x < 0$ True

$0 < x < 5$ False

$x > 5$ True

$x < 0$ True

$0 < x < 5$ False

$x > 5$ True

Step 2.9

The solution consists of all of the true intervals.

$x \leq 0$ or $x \geq 5$

Step 3

Set the denominator in $\frac{1}{\sqrt[4]{x^{2} - 5 x}}$ equal to $0$ to find where the expression is undefined.

$\sqrt[4]{x^{2} - 5 x} = 0$

Step 4

Solve for $x$ .

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Step 4.1

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $4$ .

${\sqrt[4]{x^{2} - 5 x}}^{4} = 0^{4}$

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 $\sqrt[4]{x^{2} - 5 x}$ as ${(x^{2} - 5 x)}^{\frac{1}{4}}$ .

${({(x^{2} - 5 x)}^{\frac{1}{4}})}^{4} = 0^{4}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify ${({(x^{2} - 5 x)}^{\frac{1}{4}})}^{4}$ .

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Step 4.2.2.1.1

Multiply the exponents in ${({(x^{2} - 5 x)}^{\frac{1}{4}})}^{4}$ .

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Step 4.2.2.1.1.1

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

${(x^{2} - 5 x)}^{\frac{1}{4} \cdot 4} = 0^{4}$

Step 4.2.2.1.1.2

Cancel the common factor of $4$ .

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Step 4.2.2.1.1.2.1

Cancel the common factor.

${(x^{2} - 5 x)}^{\frac{1}{4} \cdot 4} = 0^{4}$

Step 4.2.2.1.1.2.2

Rewrite the expression.

${(x^{2} - 5 x)}^{1} = 0^{4}$

Step 4.2.2.1.2

Simplify.

$x^{2} - 5 x = 0^{4}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

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

$x^{2} - 5 x = 0$

Step 4.3

Solve for $x$ .

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Step 4.3.1

Factor $x$ out of $x^{2} - 5 x$ .

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Step 4.3.1.1

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

$x \cdot x - 5 x = 0$

Step 4.3.1.2

Factor $x$ out of $- 5 x$ .

$x \cdot x + x \cdot - 5 = 0$

Step 4.3.1.3

Factor $x$ out of $x \cdot x + x \cdot - 5$ .

$x (x - 5) = 0$

Step 4.3.2

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

$x = 0$

$x - 5 = 0$

Step 4.3.3

Set $x$ equal to $0$ .

$x = 0$

Step 4.3.4

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 4.3.4.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 4.3.4.2

Add $5$ to both sides of the equation.

$x = 5$

Step 4.3.5

The final solution is all the values that make $x (x - 5) = 0$ true.

$x = 0, 5$

Step 5

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

Interval Notation:

$(- \infty, 0) \cup (5, \infty)$

Set-Builder Notation:

${x | x < 0, x > 5}$

Step 6