Calculus Examples

$\frac{1}{{(\sqrt{1 - 4 x})}^{2}}$

Step 1

Set the radicand in $\sqrt{1 - 4 x}$ greater than or equal to $0$ to find where the expression is defined.

$1 - 4 x \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $1$ from both sides of the inequality.

$- 4 x \geq - 1$

Step 2.2

Divide each term in $- 4 x \geq - 1$ by $- 4$ and simplify.

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Step 2.2.1

Divide each term in $- 4 x \geq - 1$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} \leq \frac{- 1}{- 4}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 4$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} \leq \frac{- 1}{- 4}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 1}{- 4}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Dividing two negative values results in a positive value.

$x \leq \frac{1}{4}$

Step 3

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

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

Step 4

Solve for $x$ .

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Step 4.1

Rewrite ${\sqrt{1 - 4 x}}^{2}$ as $1 - 4 x$ .

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Step 4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 - 4 x}$ as ${(1 - 4 x)}^{\frac{1}{2}}$ .

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

Step 4.1.2

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

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

Step 4.1.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 4.1.4

Cancel the common factor of $2$ .

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Step 4.1.4.1

Cancel the common factor.

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

Step 4.1.4.2

Rewrite the expression.

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

Step 4.1.5

Simplify.

$1 - 4 x = 0$

Step 4.2

Subtract $1$ from both sides of the equation.

$- 4 x = - 1$

Step 4.3

Divide each term in $- 4 x = - 1$ by $- 4$ and simplify.

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Step 4.3.1

Divide each term in $- 4 x = - 1$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 1}{- 4}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $- 4$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- 1}{- 4}$

Step 4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 4}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Dividing two negative values results in a positive value.

$x = \frac{1}{4}$

Step 5

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

Interval Notation:

$(- \infty, \frac{1}{4})$

Set-Builder Notation:

${x | x < \frac{1}{4}}$

Step 6