Trigonometry Examples

$f (x) = \frac{x - 4}{- 4 x^{2} \cdot (4 x) + 24}$

Step 1

Find the domain to determine if the expression is continuous.

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Step 1.1

Set the denominator in $\frac{x - 4}{- 4 x^{2} \cdot (4 x) + 24}$ equal to $0$ to find where the expression is undefined.

$- 4 x^{2} \cdot (4 x) + 24 = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Rewrite using the commutative property of multiplication.

$- 4 \cdot 4 x^{2} x + 24 = 0$

Step 1.2.1.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.2.1.2.1

Move $x$ .

$- 4 \cdot 4 (x \cdot x^{2}) + 24 = 0$

Step 1.2.1.2.2

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

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Step 1.2.1.2.2.1

Raise $x$ to the power of $1$ .

$- 4 \cdot 4 (x^{1} x^{2}) + 24 = 0$

Step 1.2.1.2.2.2

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

$- 4 \cdot 4 x^{1 + 2} + 24 = 0$

Step 1.2.1.2.3

Add $1$ and $2$ .

$- 4 \cdot 4 x^{3} + 24 = 0$

Step 1.2.1.3

Multiply $- 4$ by $4$ .

$- 16 x^{3} + 24 = 0$

Step 1.2.2

Subtract $24$ from both sides of the equation.

$- 16 x^{3} = - 24$

Step 1.2.3

Add $24$ to both sides of the equation.

$- 16 x^{3} + 24 = 0$

Step 1.2.4

Factor $- 8$ out of $- 16 x^{3} + 24$ .

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Step 1.2.4.1

Factor $- 8$ out of $- 16 x^{3}$ .

$- 8 (2 x^{3}) + 24 = 0$

Step 1.2.4.2

Factor $- 8$ out of $24$ .

$- 8 (2 x^{3}) - 8 \cdot - 3 = 0$

Step 1.2.4.3

Factor $- 8$ out of $- 8 (2 x^{3}) - 8 (- 3)$ .

$- 8 (2 x^{3} - 3) = 0$

Step 1.2.5

Divide each term in $- 8 (2 x^{3} - 3) = 0$ by $- 8$ and simplify.

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Step 1.2.5.1

Divide each term in $- 8 (2 x^{3} - 3) = 0$ by $- 8$ .

$\frac{- 8 (2 x^{3} - 3)}{- 8} = \frac{0}{- 8}$

Step 1.2.5.2

Simplify the left side.

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Step 1.2.5.2.1

Cancel the common factor of $- 8$ .

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Step 1.2.5.2.1.1

Cancel the common factor.

$\frac{- 8 (2 x^{3} - 3)}{- 8} = \frac{0}{- 8}$

Step 1.2.5.2.1.2

Divide $2 x^{3} - 3$ by $1$ .

$2 x^{3} - 3 = \frac{0}{- 8}$

Step 1.2.5.3

Simplify the right side.

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Step 1.2.5.3.1

Divide $0$ by $- 8$ .

$2 x^{3} - 3 = 0$

Step 1.2.6

Add $3$ to both sides of the equation.

$2 x^{3} = 3$

Step 1.2.7

Divide each term in $2 x^{3} = 3$ by $2$ and simplify.

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Step 1.2.7.1

Divide each term in $2 x^{3} = 3$ by $2$ .

$\frac{2 x^{3}}{2} = \frac{3}{2}$

Step 1.2.7.2

Simplify the left side.

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Step 1.2.7.2.1

Cancel the common factor of $2$ .

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Step 1.2.7.2.1.1

Cancel the common factor.

$\frac{2 x^{3}}{2} = \frac{3}{2}$

Step 1.2.7.2.1.2

Divide $x^{3}$ by $1$ .

$x^{3} = \frac{3}{2}$

Step 1.2.8

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

$x = \sqrt[3]{\frac{3}{2}}$

Step 1.2.9

Simplify $\sqrt[3]{\frac{3}{2}}$ .

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Step 1.2.9.1

Rewrite $\sqrt[3]{\frac{3}{2}}$ as $\frac{\sqrt[3]{3}}{\sqrt[3]{2}}$ .

$x = \frac{\sqrt[3]{3}}{\sqrt[3]{2}}$

Step 1.2.9.2

Multiply $\frac{\sqrt[3]{3}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$x = \frac{\sqrt[3]{3}}{\sqrt[3]{2}} \cdot \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$

Step 1.2.9.3

Combine and simplify the denominator.

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Step 1.2.9.3.1

Multiply $\frac{\sqrt[3]{3}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$x = \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{\sqrt[3]{2} {\sqrt[3]{2}}^{2}}$

Step 1.2.9.3.2

Raise $\sqrt[3]{2}$ to the power of $1$ .

$x = \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1} {\sqrt[3]{2}}^{2}}$

Step 1.2.9.3.3

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

$x = \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1 + 2}}$

Step 1.2.9.3.4

Add $1$ and $2$ .

$x = \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{3}}$

Step 1.2.9.3.5

Rewrite ${\sqrt[3]{2}}^{3}$ as $2$ .

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Step 1.2.9.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2}$ as $2^{\frac{1}{3}}$ .

$x = \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{{(2^{\frac{1}{3}})}^{3}}$

Step 1.2.9.3.5.2

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

$x = \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{2^{\frac{1}{3} \cdot 3}}$

Step 1.2.9.3.5.3

Combine $\frac{1}{3}$ and $3$ .

$x = \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 1.2.9.3.5.4

Cancel the common factor of $3$ .

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Step 1.2.9.3.5.4.1

Cancel the common factor.

$x = \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 1.2.9.3.5.4.2

Rewrite the expression.

$x = \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{2^{1}}$

Step 1.2.9.3.5.5

Evaluate the exponent.

$x = \frac{\sqrt[3]{3} {\sqrt[3]{2}}^{2}}{2}$

Step 1.2.9.4

Simplify the numerator.

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Step 1.2.9.4.1

Rewrite ${\sqrt[3]{2}}^{2}$ as $\sqrt[3]{2^{2}}$ .

$x = \frac{\sqrt[3]{3} \sqrt[3]{2^{2}}}{2}$

Step 1.2.9.4.2

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

$x = \frac{\sqrt[3]{3} \sqrt[3]{4}}{2}$

Step 1.2.9.5

Simplify the numerator.

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Step 1.2.9.5.1

Combine using the product rule for radicals.

$x = \frac{\sqrt[3]{3 \cdot 4}}{2}$

Step 1.2.9.5.2

Multiply $3$ by $4$ .

$x = \frac{\sqrt[3]{12}}{2}$

Step 1.3

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

Interval Notation:

$(- \infty, \frac{\sqrt[3]{12}}{2}) \cup (\frac{\sqrt[3]{12}}{2}, \infty)$

Set-Builder Notation:

${x | x \neq \frac{\sqrt[3]{12}}{2}}$

Interval Notation:

$(- \infty, \frac{\sqrt[3]{12}}{2}) \cup (\frac{\sqrt[3]{12}}{2}, \infty)$

Set-Builder Notation:

${x | x \neq \frac{\sqrt[3]{12}}{2}}$

Step 2

Since the domain is not all real numbers, $\frac{x - 4}{- 4 x^{2} \cdot (4 x) + 24}$ is not continuous over all real numbers.

Not continuous

Step 3