Algebra Examples

$y = 7 x^{3} + 2$

Step 1

Interchange the variables.

$x = 7 y^{3} + 2$

Step 2

Solve for $y$ .

Tap for more steps...

Step 2.1

Rewrite the equation as $7 y^{3} + 2 = x$ .

$7 y^{3} + 2 = x$

Step 2.2

Subtract $2$ from both sides of the equation.

$7 y^{3} = x - 2$

Step 2.3

Divide each term in $7 y^{3} = x - 2$ by $7$ and simplify.

Tap for more steps...

Step 2.3.1

Divide each term in $7 y^{3} = x - 2$ by $7$ .

$\frac{7 y^{3}}{7} = \frac{x}{7} + \frac{- 2}{7}$

Step 2.3.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.1

Cancel the common factor of $7$ .

Tap for more steps...

Step 2.3.2.1.1

Cancel the common factor.

$\frac{7 y^{3}}{7} = \frac{x}{7} + \frac{- 2}{7}$

Step 2.3.2.1.2

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

$y^{3} = \frac{x}{7} + \frac{- 2}{7}$

Step 2.3.3

Simplify the right side.

Tap for more steps...

Step 2.3.3.1

Move the negative in front of the fraction.

$y^{3} = \frac{x}{7} - \frac{2}{7}$

Step 2.4

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

$y = \sqrt[3]{\frac{x}{7} - \frac{2}{7}}$

Step 2.5

Simplify $\sqrt[3]{\frac{x}{7} - \frac{2}{7}}$ .

Tap for more steps...

Step 2.5.1

Combine the numerators over the common denominator.

$y = \sqrt[3]{\frac{x - 2}{7}}$

Step 2.5.2

Rewrite $\sqrt[3]{\frac{x - 2}{7}}$ as $\frac{\sqrt[3]{x - 2}}{\sqrt[3]{7}}$ .

$y = \frac{\sqrt[3]{x - 2}}{\sqrt[3]{7}}$

Step 2.5.3

Multiply $\frac{\sqrt[3]{x - 2}}{\sqrt[3]{7}}$ by $\frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$ .

$y = \frac{\sqrt[3]{x - 2}}{\sqrt[3]{7}} \cdot \frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$

Step 2.5.4

Combine and simplify the denominator.

Tap for more steps...

Step 2.5.4.1

Multiply $\frac{\sqrt[3]{x - 2}}{\sqrt[3]{7}}$ by $\frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$ .

$y = \frac{\sqrt[3]{x - 2} {\sqrt[3]{7}}^{2}}{\sqrt[3]{7} {\sqrt[3]{7}}^{2}}$

Step 2.5.4.2

Raise $\sqrt[3]{7}$ to the power of $1$ .

$y = \frac{\sqrt[3]{x - 2} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{1} {\sqrt[3]{7}}^{2}}$

Step 2.5.4.3

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

$y = \frac{\sqrt[3]{x - 2} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{1 + 2}}$

Step 2.5.4.4

Add $1$ and $2$ .

$y = \frac{\sqrt[3]{x - 2} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{3}}$

Step 2.5.4.5

Rewrite ${\sqrt[3]{7}}^{3}$ as $7$ .

Tap for more steps...

Step 2.5.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{7}$ as $7^{\frac{1}{3}}$ .

$y = \frac{\sqrt[3]{x - 2} {\sqrt[3]{7}}^{2}}{{(7^{\frac{1}{3}})}^{3}}$

Step 2.5.4.5.2

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

$y = \frac{\sqrt[3]{x - 2} {\sqrt[3]{7}}^{2}}{7^{\frac{1}{3} \cdot 3}}$

Step 2.5.4.5.3

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

$y = \frac{\sqrt[3]{x - 2} {\sqrt[3]{7}}^{2}}{7^{\frac{3}{3}}}$

Step 2.5.4.5.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.5.4.5.4.1

Cancel the common factor.

$y = \frac{\sqrt[3]{x - 2} {\sqrt[3]{7}}^{2}}{7^{\frac{3}{3}}}$

Step 2.5.4.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[3]{x - 2} {\sqrt[3]{7}}^{2}}{7^{1}}$

Step 2.5.4.5.5

Evaluate the exponent.

$y = \frac{\sqrt[3]{x - 2} {\sqrt[3]{7}}^{2}}{7}$

Step 2.5.5

Simplify the numerator.

Tap for more steps...

Step 2.5.5.1

Rewrite ${\sqrt[3]{7}}^{2}$ as $\sqrt[3]{7^{2}}$ .

$y = \frac{\sqrt[3]{x - 2} \sqrt[3]{7^{2}}}{7}$

Step 2.5.5.2

Raise $7$ to the power of $2$ .

$y = \frac{\sqrt[3]{x - 2} \sqrt[3]{49}}{7}$

Step 2.5.6

Simplify with factoring out.

Tap for more steps...

Step 2.5.6.1

Combine using the product rule for radicals.

$y = \frac{\sqrt[3]{(x - 2) \cdot 49}}{7}$

Step 2.5.6.2

Reorder factors in $\frac{\sqrt[3]{(x - 2) \cdot 49}}{7}$ .

$y = \frac{\sqrt[3]{49 (x - 2)}}{7}$

Step 3

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = \frac{\sqrt[3]{49 (x - 2)}}{7}$

Step 4

Verify if $f^{- 1} (x) = \frac{\sqrt[3]{49 (x - 2)}}{7}$ is the inverse of $f (x) = 7 x^{3} + 2$ .

Tap for more steps...

Step 4.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 4.2

Evaluate $f^{- 1} (f (x))$ .

Tap for more steps...

Step 4.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 4.2.2

Evaluate $f^{- 1} (7 x^{3} + 2)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (7 x^{3} + 2) = \frac{\sqrt[3]{49 ((7 x^{3} + 2) - 2)}}{7}$

Step 4.2.3

Simplify the numerator.

Tap for more steps...

Step 4.2.3.1

Subtract $2$ from $2$ .

$f^{- 1} (7 x^{3} + 2) = \frac{\sqrt[3]{49 (7 x^{3} + 0)}}{7}$

Step 4.2.3.2

Add $7 x^{3}$ and $0$ .

$f^{- 1} (7 x^{3} + 2) = \frac{\sqrt[3]{49 \cdot (7 x^{3})}}{7}$

Step 4.2.3.3

Multiply $49$ by $7$ .

$f^{- 1} (7 x^{3} + 2) = \frac{\sqrt[3]{343 x^{3}}}{7}$

Step 4.2.3.4

Rewrite $343 x^{3}$ as ${(7 x)}^{3}$ .

$f^{- 1} (7 x^{3} + 2) = \frac{\sqrt[3]{{(7 x)}^{3}}}{7}$

Step 4.2.3.5

Pull terms out from under the radical, assuming real numbers.

$f^{- 1} (7 x^{3} + 2) = \frac{7 x}{7}$

Step 4.2.4

Cancel the common factor of $7$ .

Tap for more steps...

Step 4.2.4.1

Cancel the common factor.

$f^{- 1} (7 x^{3} + 2) = \frac{7 x}{7}$

Step 4.2.4.2

Divide $x$ by $1$ .

$f^{- 1} (7 x^{3} + 2) = x$

Step 4.3

Evaluate $f (f^{- 1} (x))$ .

Tap for more steps...

Step 4.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 4.3.2

Evaluate $f (\frac{\sqrt[3]{49 (x - 2)}}{7})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\sqrt[3]{49 (x - 2)}}{7}) = 7 {(\frac{\sqrt[3]{49 (x - 2)}}{7})}^{3} + 2$

Step 4.3.3

Simplify each term.

Tap for more steps...

Step 4.3.3.1

Apply the product rule to $\frac{\sqrt[3]{49 (x - 2)}}{7}$ .

$f (\frac{\sqrt[3]{49 (x - 2)}}{7}) = 7 (\frac{{\sqrt[3]{49 (x - 2)}}^{3}}{7^{3}}) + 2$

Step 4.3.3.2

Simplify the numerator.

Tap for more steps...

Step 4.3.3.2.1

Rewrite ${\sqrt[3]{49 (x - 2)}}^{3}$ as $49 (x - 2)$ .

Tap for more steps...

Step 4.3.3.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{49 (x - 2)}$ as ${(49 (x - 2))}^{\frac{1}{3}}$ .

$f (\frac{\sqrt[3]{49 (x - 2)}}{7}) = 7 (\frac{{({(49 (x - 2))}^{\frac{1}{3}})}^{3}}{7^{3}}) + 2$

Step 4.3.3.2.1.2

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

$f (\frac{\sqrt[3]{49 (x - 2)}}{7}) = 7 (\frac{{(49 (x - 2))}^{\frac{1}{3} \cdot 3}}{7^{3}}) + 2$

Step 4.3.3.2.1.3

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

$f (\frac{\sqrt[3]{49 (x - 2)}}{7}) = 7 (\frac{{(49 (x - 2))}^{\frac{3}{3}}}{7^{3}}) + 2$

Step 4.3.3.2.1.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.3.3.2.1.4.1

Cancel the common factor.

$f (\frac{\sqrt[3]{49 (x - 2)}}{7}) = 7 (\frac{{(49 (x - 2))}^{\frac{3}{3}}}{7^{3}}) + 2$

Step 4.3.3.2.1.4.2

Rewrite the expression.

$f (\frac{\sqrt[3]{49 (x - 2)}}{7}) = 7 (\frac{49 (x - 2)}{7^{3}}) + 2$

Step 4.3.3.2.1.5

Simplify.

$f (\frac{\sqrt[3]{49 (x - 2)}}{7}) = 7 (\frac{49 (x - 2)}{7^{3}}) + 2$

Step 4.3.3.2.2

Apply the distributive property.

$f (\frac{\sqrt[3]{49 (x - 2)}}{7}) = 7 (\frac{49 x + 49 \cdot - 2}{7^{3}}) + 2$

Step 4.3.3.2.3

Multiply $49$ by $- 2$ .

$f (\frac{\sqrt[3]{49 (x - 2)}}{7}) = 7 (\frac{49 x - 98}{7^{3}}) + 2$

Step 4.3.3.2.4

Factor $49$ out of $49 x - 98$ .

Tap for more steps...

Step 4.3.3.2.4.1

Factor $49$ out of $49 x$ .

$f (\frac{\sqrt[3]{49 (x - 2)}}{7}) = 7 (\frac{49 (x) - 98}{7^{3}}) + 2$

Step 4.3.3.2.4.2

Factor $49$ out of $- 98$ .

$f (\frac{\sqrt[3]{49 (x - 2)}}{7}) = 7 (\frac{49 x + 49 \cdot - 2}{7^{3}}) + 2$

Step 4.3.3.2.4.3

Factor $49$ out of $49 x + 49 \cdot - 2$ .

$f (\frac{\sqrt[3]{49 (x - 2)}}{7}) = 7 (\frac{49 (x - 2)}{7^{3}}) + 2$

Step 4.3.3.3

Raise $7$ to the power of $3$ .

$f (\frac{\sqrt[3]{49 (x - 2)}}{7}) = 7 (\frac{49 (x - 2)}{343}) + 2$

Step 4.3.3.4

Cancel the common factor of $7$ .

Tap for more steps...

Step 4.3.3.4.1

Factor $7$ out of $343$ .

$f (\frac{\sqrt[3]{49 (x - 2)}}{7}) = 7 (\frac{49 (x - 2)}{7 (49)}) + 2$

Step 4.3.3.4.2

Cancel the common factor.

$f (\frac{\sqrt[3]{49 (x - 2)}}{7}) = 7 (\frac{49 (x - 2)}{7 \cdot 49}) + 2$

Step 4.3.3.4.3

Rewrite the expression.

$f (\frac{\sqrt[3]{49 (x - 2)}}{7}) = \frac{49 (x - 2)}{49} + 2$

Step 4.3.3.5

Cancel the common factor of $49$ .

Tap for more steps...

Step 4.3.3.5.1

Cancel the common factor.

$f (\frac{\sqrt[3]{49 (x - 2)}}{7}) = \frac{49 (x - 2)}{49} + 2$

Step 4.3.3.5.2

Divide $x - 2$ by $1$ .

$f (\frac{\sqrt[3]{49 (x - 2)}}{7}) = x - 2 + 2$

Step 4.3.4

Combine the opposite terms in $x - 2 + 2$ .

Tap for more steps...

Step 4.3.4.1

Add $- 2$ and $2$ .

$f (\frac{\sqrt[3]{49 (x - 2)}}{7}) = x + 0$

Step 4.3.4.2

Add $x$ and $0$ .

$f (\frac{\sqrt[3]{49 (x - 2)}}{7}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\sqrt[3]{49 (x - 2)}}{7}$ is the inverse of $f (x) = 7 x^{3} + 2$ .

$f^{- 1} (x) = \frac{\sqrt[3]{49 (x - 2)}}{7}$