Calculus Examples

$f (x) = x - 9 x^{\frac{1}{3}}$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = x - 9 x^{\frac{1}{3}}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $x - 9 x^{\frac{1}{3}} = 0$ .

$x - 9 x^{\frac{1}{3}} = 0$

Step 1.2.2

Find a common factor $x^{\frac{1}{3}}$ that is present in each term.

${(x^{\frac{1}{3}})}^{3} - 9 x^{\frac{1}{3}}$

Step 1.2.3

Substitute $u$ for $x^{\frac{1}{3}}$ .

${(u)}^{3} - 9 u = 0$

Step 1.2.4

Solve for $u$ .

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Step 1.2.4.1

Remove parentheses.

$u^{3} - 9 u = 0$

Step 1.2.4.2

Factor the left side of the equation.

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Step 1.2.4.2.1

Factor $u$ out of $u^{3} - 9 u$ .

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Step 1.2.4.2.1.1

Factor $u$ out of $u^{3}$ .

$u \cdot u^{2} - 9 u = 0$

Step 1.2.4.2.1.2

Factor $u$ out of $- 9 u$ .

$u \cdot u^{2} + u \cdot - 9 = 0$

Step 1.2.4.2.1.3

Factor $u$ out of $u \cdot u^{2} + u \cdot - 9$ .

$u (u^{2} - 9) = 0$

Step 1.2.4.2.2

Rewrite $9$ as $3^{2}$ .

$u (u^{2} - 3^{2}) = 0$

Step 1.2.4.2.3

Factor.

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Step 1.2.4.2.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = u$ and $b = 3$ .

$u ((u + 3) (u - 3)) = 0$

Step 1.2.4.2.3.2

Remove unnecessary parentheses.

$u (u + 3) (u - 3) = 0$

Step 1.2.4.3

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

$u = 0$

$u + 3 = 0$

$u - 3 = 0$

Step 1.2.4.4

Set $u$ equal to $0$ .

$u = 0$

Step 1.2.4.5

Set $u + 3$ equal to $0$ and solve for $u$ .

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Step 1.2.4.5.1

Set $u + 3$ equal to $0$ .

$u + 3 = 0$

Step 1.2.4.5.2

Subtract $3$ from both sides of the equation.

$u = - 3$

Step 1.2.4.6

Set $u - 3$ equal to $0$ and solve for $u$ .

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Step 1.2.4.6.1

Set $u - 3$ equal to $0$ .

$u - 3 = 0$

Step 1.2.4.6.2

Add $3$ to both sides of the equation.

$u = 3$

Step 1.2.4.7

The final solution is all the values that make $u (u + 3) (u - 3) = 0$ true.

$u = 0, - 3, 3$

Step 1.2.5

Substitute $x$ for $u$ .

$x^{\frac{1}{3}} = 0, - 3, 3$

Step 1.2.6

Solve for $x^{\frac{1}{3}} = 0$ for $x$ .

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Step 1.2.6.1

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{3}})}^{3} = 0^{3}$

Step 1.2.6.2

Simplify the exponent.

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Step 1.2.6.2.1

Simplify the left side.

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Step 1.2.6.2.1.1

Simplify ${(x^{\frac{1}{3}})}^{3}$ .

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Step 1.2.6.2.1.1.1

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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Step 1.2.6.2.1.1.1.1

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

$x^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 1.2.6.2.1.1.1.2

Cancel the common factor of $3$ .

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Step 1.2.6.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 1.2.6.2.1.1.1.2.2

Rewrite the expression.

$x^{1} = 0^{3}$

Step 1.2.6.2.1.1.2

Simplify.

$x = 0^{3}$

Step 1.2.6.2.2

Simplify the right side.

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Step 1.2.6.2.2.1

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

$x = 0$

Step 1.2.7

Solve for $x^{\frac{1}{3}} = - 3$ for $x$ .

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Step 1.2.7.1

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{3}})}^{3} = {(- 3)}^{3}$

Step 1.2.7.2

Simplify the exponent.

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Step 1.2.7.2.1

Simplify the left side.

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Step 1.2.7.2.1.1

Simplify ${(x^{\frac{1}{3}})}^{3}$ .

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Step 1.2.7.2.1.1.1

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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Step 1.2.7.2.1.1.1.1

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

$x^{\frac{1}{3} \cdot 3} = {(- 3)}^{3}$

Step 1.2.7.2.1.1.1.2

Cancel the common factor of $3$ .

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Step 1.2.7.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = {(- 3)}^{3}$

Step 1.2.7.2.1.1.1.2.2

Rewrite the expression.

$x^{1} = {(- 3)}^{3}$

Step 1.2.7.2.1.1.2

Simplify.

$x = {(- 3)}^{3}$

Step 1.2.7.2.2

Simplify the right side.

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Step 1.2.7.2.2.1

Raise $- 3$ to the power of $3$ .

$x = - 27$

Step 1.2.8

Solve for $x^{\frac{1}{3}} = 3$ for $x$ .

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Step 1.2.8.1

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{3}})}^{3} = 3^{3}$

Step 1.2.8.2

Simplify the exponent.

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Step 1.2.8.2.1

Simplify the left side.

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Step 1.2.8.2.1.1

Simplify ${(x^{\frac{1}{3}})}^{3}$ .

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Step 1.2.8.2.1.1.1

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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Step 1.2.8.2.1.1.1.1

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

$x^{\frac{1}{3} \cdot 3} = 3^{3}$

Step 1.2.8.2.1.1.1.2

Cancel the common factor of $3$ .

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Step 1.2.8.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = 3^{3}$

Step 1.2.8.2.1.1.1.2.2

Rewrite the expression.

$x^{1} = 3^{3}$

Step 1.2.8.2.1.1.2

Simplify.

$x = 3^{3}$

Step 1.2.8.2.2

Simplify the right side.

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Step 1.2.8.2.2.1

Raise $3$ to the power of $3$ .

$x = 27$

Step 1.2.9

List all of the solutions.

$x = 0, - 27, 27$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- 27, 0), (27, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = (0) - 9 {(0)}^{\frac{1}{3}}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 0 - 9 \cdot 0^{\frac{1}{3}}$

Step 2.2.2

Remove parentheses.

$y = (0) - 9 {(0)}^{\frac{1}{3}}$

Step 2.2.3

Simplify $(0) - 9 {(0)}^{\frac{1}{3}}$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Rewrite $0$ as $0^{3}$ .

$y = 0 - 9 {(0^{3})}^{\frac{1}{3}}$

Step 2.2.3.1.2

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

$y = 0 - 9 \cdot 0^{3 (\frac{1}{3})}$

Step 2.2.3.1.3

Cancel the common factor of $3$ .

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Step 2.2.3.1.3.1

Cancel the common factor.

$y = 0 - 9 \cdot 0^{3 (\frac{1}{3})}$

Step 2.2.3.1.3.2

Rewrite the expression.

$y = 0 - 9 \cdot 0^{1}$

Step 2.2.3.1.4

Evaluate the exponent.

$y = 0 - 9 \cdot 0$

Step 2.2.3.1.5

Multiply $- 9$ by $0$ .

$y = 0 + 0$

Step 2.2.3.2

Add $0$ and $0$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 0)$

Step 3

List the intersections.

x-intercept(s): $(0, 0), (- 27, 0), (27, 0)$

y-intercept(s): $(0, 0)$

Step 4