Calculus Examples

$h (x) = \frac{1}{\sqrt{7 x^{2} + 6}}$

Step 1

Find the domain to determine if the expression is continuous.

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Step 1.1

Set the radicand in $\sqrt{7 x^{2} + 6}$ greater than or equal to $0$ to find where the expression is defined.

$7 x^{2} + 6 \geq 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Subtract $6$ from both sides of the inequality.

$7 x^{2} \geq - 6$

Step 1.2.2

Divide each term in $7 x^{2} \geq - 6$ by $7$ and simplify.

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Step 1.2.2.1

Divide each term in $7 x^{2} \geq - 6$ by $7$ .

$\frac{7 x^{2}}{7} \geq \frac{- 6}{7}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $7$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{7 x^{2}}{7} \geq \frac{- 6}{7}$

Step 1.2.2.2.1.2

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

$x^{2} \geq \frac{- 6}{7}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Move the negative in front of the fraction.

$x^{2} \geq - \frac{6}{7}$

Step 1.2.3

Since the left side has an even power, it is always positive for all real numbers.

All real numbers

Step 1.3

Set the denominator in $\frac{1}{\sqrt{7 x^{2} + 6}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{7 x^{2} + 6} = 0$

Step 1.4

Solve for $x$ .

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Step 1.4.1

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

${\sqrt{7 x^{2} + 6}}^{2} = 0^{2}$

Step 1.4.2

Simplify each side of the equation.

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Step 1.4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 x^{2} + 6}$ as ${(7 x^{2} + 6)}^{\frac{1}{2}}$ .

${({(7 x^{2} + 6)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 1.4.2.2

Simplify the left side.

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Step 1.4.2.2.1

Simplify ${({(7 x^{2} + 6)}^{\frac{1}{2}})}^{2}$ .

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Step 1.4.2.2.1.1

Multiply the exponents in ${({(7 x^{2} + 6)}^{\frac{1}{2}})}^{2}$ .

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Step 1.4.2.2.1.1.1

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

${(7 x^{2} + 6)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.4.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 1.4.2.2.1.1.2.1

Cancel the common factor.

${(7 x^{2} + 6)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.4.2.2.1.1.2.2

Rewrite the expression.

${(7 x^{2} + 6)}^{1} = 0^{2}$

Step 1.4.2.2.1.2

Simplify.

$7 x^{2} + 6 = 0^{2}$

Step 1.4.2.3

Simplify the right side.

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Step 1.4.2.3.1

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

$7 x^{2} + 6 = 0$

Step 1.4.3

Solve for $x$ .

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Step 1.4.3.1

Subtract $6$ from both sides of the equation.

$7 x^{2} = - 6$

Step 1.4.3.2

Divide each term in $7 x^{2} = - 6$ by $7$ and simplify.

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Step 1.4.3.2.1

Divide each term in $7 x^{2} = - 6$ by $7$ .

$\frac{7 x^{2}}{7} = \frac{- 6}{7}$

Step 1.4.3.2.2

Simplify the left side.

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Step 1.4.3.2.2.1

Cancel the common factor of $7$ .

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Step 1.4.3.2.2.1.1

Cancel the common factor.

$\frac{7 x^{2}}{7} = \frac{- 6}{7}$

Step 1.4.3.2.2.1.2

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

$x^{2} = \frac{- 6}{7}$

Step 1.4.3.2.3

Simplify the right side.

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Step 1.4.3.2.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{6}{7}$

Step 1.4.3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- \frac{6}{7}}$

Step 1.4.3.4

Simplify $\pm \sqrt{- \frac{6}{7}}$ .

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Step 1.4.3.4.1

Rewrite $- 1$ as $i^{2}$ .

$x = \pm \sqrt{i^{2} \frac{6}{7}}$

Step 1.4.3.4.2

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{6}{7}}$

Step 1.4.3.4.3

Rewrite $\sqrt{\frac{6}{7}}$ as $\frac{\sqrt{6}}{\sqrt{7}}$ .

$x = \pm i \frac{\sqrt{6}}{\sqrt{7}}$

Step 1.4.3.4.4

Multiply $\frac{\sqrt{6}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x = \pm i (\frac{\sqrt{6}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}})$

Step 1.4.3.4.5

Combine and simplify the denominator.

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Step 1.4.3.4.5.1

Multiply $\frac{\sqrt{6}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x = \pm i \frac{\sqrt{6} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 1.4.3.4.5.2

Raise $\sqrt{7}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{6} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 1.4.3.4.5.3

Raise $\sqrt{7}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{6} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 1.4.3.4.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm i \frac{\sqrt{6} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 1.4.3.4.5.5

Add $1$ and $1$ .

$x = \pm i \frac{\sqrt{6} \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 1.4.3.4.5.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 1.4.3.4.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$x = \pm i \frac{\sqrt{6} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 1.4.3.4.5.6.2

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

$x = \pm i \frac{\sqrt{6} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 1.4.3.4.5.6.3

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

$x = \pm i \frac{\sqrt{6} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 1.4.3.4.5.6.4

Cancel the common factor of $2$ .

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Step 1.4.3.4.5.6.4.1

Cancel the common factor.

$x = \pm i \frac{\sqrt{6} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 1.4.3.4.5.6.4.2

Rewrite the expression.

$x = \pm i \frac{\sqrt{6} \sqrt{7}}{7^{1}}$

Step 1.4.3.4.5.6.5

Evaluate the exponent.

$x = \pm i \frac{\sqrt{6} \sqrt{7}}{7}$

Step 1.4.3.4.6

Simplify the numerator.

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Step 1.4.3.4.6.1

Combine using the product rule for radicals.

$x = \pm i \frac{\sqrt{6 \cdot 7}}{7}$

Step 1.4.3.4.6.2

Multiply $6$ by $7$ .

$x = \pm i \frac{\sqrt{42}}{7}$

Step 1.4.3.4.7

Combine $i$ and $\frac{\sqrt{42}}{7}$ .

$x = \pm \frac{i \sqrt{42}}{7}$

Step 1.4.3.5

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

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Step 1.4.3.5.1

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

$x = \frac{i \sqrt{42}}{7}$

Step 1.4.3.5.2

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

$x = - \frac{i \sqrt{42}}{7}$

Step 1.4.3.5.3

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

$x = \frac{i \sqrt{42}}{7}, - \frac{i \sqrt{42}}{7}$

Step 1.5

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 2

Since the domain is all real numbers, $\frac{1}{\sqrt{7 x^{2} + 6}}$ is continuous over all real numbers.

Continuous

Step 3