Precalculus Examples

$f (x) = 5 x^{3} + 5$ ; find $f^{- 1} (x)$

Step 1

Write $f (x) = 5 x^{3} + 5$ as an equation.

$y = 5 x^{3} + 5$

Step 2

Interchange the variables.

$x = 5 y^{3} + 5$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $5 y^{3} + 5 = x$ .

$5 y^{3} + 5 = x$

Step 3.2

Subtract $5$ from both sides of the equation.

$5 y^{3} = x - 5$

Step 3.3

Divide each term in $5 y^{3} = x - 5$ by $5$ and simplify.

Tap for more steps...

Step 3.3.1

Divide each term in $5 y^{3} = x - 5$ by $5$ .

$\frac{5 y^{3}}{5} = \frac{x}{5} + \frac{- 5}{5}$

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 3.3.2.1.1

Cancel the common factor.

$\frac{5 y^{3}}{5} = \frac{x}{5} + \frac{- 5}{5}$

Step 3.3.2.1.2

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

$y^{3} = \frac{x}{5} + \frac{- 5}{5}$

Step 3.3.3

Simplify the right side.

Tap for more steps...

Step 3.3.3.1

Divide $- 5$ by $5$ .

$y^{3} = \frac{x}{5} - 1$

Step 3.4

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

$y = \sqrt[3]{\frac{x}{5} - 1}$

Step 3.5

Simplify $\sqrt[3]{\frac{x}{5} - 1}$ .

Tap for more steps...

Step 3.5.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \sqrt[3]{\frac{x}{5} - 1 \cdot \frac{5}{5}}$

Step 3.5.2

Combine $- 1$ and $\frac{5}{5}$ .

$y = \sqrt[3]{\frac{x}{5} + \frac{- 1 \cdot 5}{5}}$

Step 3.5.3

Combine the numerators over the common denominator.

$y = \sqrt[3]{\frac{x - 1 \cdot 5}{5}}$

Step 3.5.4

Multiply $- 1$ by $5$ .

$y = \sqrt[3]{\frac{x - 5}{5}}$

Step 3.5.5

Rewrite $\sqrt[3]{\frac{x - 5}{5}}$ as $\frac{\sqrt[3]{x - 5}}{\sqrt[3]{5}}$ .

$y = \frac{\sqrt[3]{x - 5}}{\sqrt[3]{5}}$

Step 3.5.6

Multiply $\frac{\sqrt[3]{x - 5}}{\sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

$y = \frac{\sqrt[3]{x - 5}}{\sqrt[3]{5}} \cdot \frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$

Step 3.5.7

Combine and simplify the denominator.

Tap for more steps...

Step 3.5.7.1

Multiply $\frac{\sqrt[3]{x - 5}}{\sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

$y = \frac{\sqrt[3]{x - 5} {\sqrt[3]{5}}^{2}}{\sqrt[3]{5} {\sqrt[3]{5}}^{2}}$

Step 3.5.7.2

Raise $\sqrt[3]{5}$ to the power of $1$ .

$y = \frac{\sqrt[3]{x - 5} {\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{1} {\sqrt[3]{5}}^{2}}$

Step 3.5.7.3

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

$y = \frac{\sqrt[3]{x - 5} {\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{1 + 2}}$

Step 3.5.7.4

Add $1$ and $2$ .

$y = \frac{\sqrt[3]{x - 5} {\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{3}}$

Step 3.5.7.5

Rewrite ${\sqrt[3]{5}}^{3}$ as $5$ .

Tap for more steps...

Step 3.5.7.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5}$ as $5^{\frac{1}{3}}$ .

$y = \frac{\sqrt[3]{x - 5} {\sqrt[3]{5}}^{2}}{{(5^{\frac{1}{3}})}^{3}}$

Step 3.5.7.5.2

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

$y = \frac{\sqrt[3]{x - 5} {\sqrt[3]{5}}^{2}}{5^{\frac{1}{3} \cdot 3}}$

Step 3.5.7.5.3

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

$y = \frac{\sqrt[3]{x - 5} {\sqrt[3]{5}}^{2}}{5^{\frac{3}{3}}}$

Step 3.5.7.5.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.5.7.5.4.1

Cancel the common factor.

$y = \frac{\sqrt[3]{x - 5} {\sqrt[3]{5}}^{2}}{5^{\frac{3}{3}}}$

Step 3.5.7.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[3]{x - 5} {\sqrt[3]{5}}^{2}}{5^{1}}$

Step 3.5.7.5.5

Evaluate the exponent.

$y = \frac{\sqrt[3]{x - 5} {\sqrt[3]{5}}^{2}}{5}$

Step 3.5.8

Simplify the numerator.

Tap for more steps...

Step 3.5.8.1

Rewrite ${\sqrt[3]{5}}^{2}$ as $\sqrt[3]{5^{2}}$ .

$y = \frac{\sqrt[3]{x - 5} \sqrt[3]{5^{2}}}{5}$

Step 3.5.8.2

Raise $5$ to the power of $2$ .

$y = \frac{\sqrt[3]{x - 5} \sqrt[3]{25}}{5}$

Step 3.5.9

Simplify with factoring out.

Tap for more steps...

Step 3.5.9.1

Combine using the product rule for radicals.

$y = \frac{\sqrt[3]{(x - 5) \cdot 25}}{5}$

Step 3.5.9.2

Reorder factors in $\frac{\sqrt[3]{(x - 5) \cdot 25}}{5}$ .

$y = \frac{\sqrt[3]{25 (x - 5)}}{5}$

Step 4

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

$f^{- 1} (x) = \frac{\sqrt[3]{25 (x - 5)}}{5}$

Step 5

Verify if $f^{- 1} (x) = \frac{\sqrt[3]{25 (x - 5)}}{5}$ is the inverse of $f (x) = 5 x^{3} + 5$ .

Tap for more steps...

Step 5.1

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (5 x^{3} + 5)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (5 x^{3} + 5) = \frac{\sqrt[3]{25 ((5 x^{3} + 5) - 5)}}{5}$

Step 5.2.3

Simplify the numerator.

Tap for more steps...

Step 5.2.3.1

Subtract $5$ from $5$ .

$f^{- 1} (5 x^{3} + 5) = \frac{\sqrt[3]{25 (5 x^{3} + 0)}}{5}$

Step 5.2.3.2

Add $5 x^{3}$ and $0$ .

$f^{- 1} (5 x^{3} + 5) = \frac{\sqrt[3]{25 \cdot (5 x^{3})}}{5}$

Step 5.2.3.3

Multiply $25$ by $5$ .

$f^{- 1} (5 x^{3} + 5) = \frac{\sqrt[3]{125 x^{3}}}{5}$

Step 5.2.3.4

Rewrite $125 x^{3}$ as ${(5 x)}^{3}$ .

$f^{- 1} (5 x^{3} + 5) = \frac{\sqrt[3]{{(5 x)}^{3}}}{5}$

Step 5.2.3.5

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

$f^{- 1} (5 x^{3} + 5) = \frac{5 x}{5}$

Step 5.2.4

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.2.4.1

Cancel the common factor.

$f^{- 1} (5 x^{3} + 5) = \frac{5 x}{5}$

Step 5.2.4.2

Divide $x$ by $1$ .

$f^{- 1} (5 x^{3} + 5) = x$

Step 5.3

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

Tap for more steps...

Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (\frac{\sqrt[3]{25 (x - 5)}}{5})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\sqrt[3]{25 (x - 5)}}{5}) = 5 {(\frac{\sqrt[3]{25 (x - 5)}}{5})}^{3} + 5$

Step 5.3.3

Simplify each term.

Tap for more steps...

Step 5.3.3.1

Apply the product rule to $\frac{\sqrt[3]{25 (x - 5)}}{5}$ .

$f (\frac{\sqrt[3]{25 (x - 5)}}{5}) = 5 (\frac{{\sqrt[3]{25 (x - 5)}}^{3}}{5^{3}}) + 5$

Step 5.3.3.2

Simplify the numerator.

Tap for more steps...

Step 5.3.3.2.1

Rewrite ${\sqrt[3]{25 (x - 5)}}^{3}$ as $25 (x - 5)$ .

Tap for more steps...

Step 5.3.3.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{25 (x - 5)}$ as ${(25 (x - 5))}^{\frac{1}{3}}$ .

$f (\frac{\sqrt[3]{25 (x - 5)}}{5}) = 5 (\frac{{({(25 (x - 5))}^{\frac{1}{3}})}^{3}}{5^{3}}) + 5$

Step 5.3.3.2.1.2

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

$f (\frac{\sqrt[3]{25 (x - 5)}}{5}) = 5 (\frac{{(25 (x - 5))}^{\frac{1}{3} \cdot 3}}{5^{3}}) + 5$

Step 5.3.3.2.1.3

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

$f (\frac{\sqrt[3]{25 (x - 5)}}{5}) = 5 (\frac{{(25 (x - 5))}^{\frac{3}{3}}}{5^{3}}) + 5$

Step 5.3.3.2.1.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.3.3.2.1.4.1

Cancel the common factor.

$f (\frac{\sqrt[3]{25 (x - 5)}}{5}) = 5 (\frac{{(25 (x - 5))}^{\frac{3}{3}}}{5^{3}}) + 5$

Step 5.3.3.2.1.4.2

Rewrite the expression.

$f (\frac{\sqrt[3]{25 (x - 5)}}{5}) = 5 (\frac{25 (x - 5)}{5^{3}}) + 5$

Step 5.3.3.2.1.5

Simplify.

$f (\frac{\sqrt[3]{25 (x - 5)}}{5}) = 5 (\frac{25 (x - 5)}{5^{3}}) + 5$

Step 5.3.3.2.2

Apply the distributive property.

$f (\frac{\sqrt[3]{25 (x - 5)}}{5}) = 5 (\frac{25 x + 25 \cdot - 5}{5^{3}}) + 5$

Step 5.3.3.2.3

Multiply $25$ by $- 5$ .

$f (\frac{\sqrt[3]{25 (x - 5)}}{5}) = 5 (\frac{25 x - 125}{5^{3}}) + 5$

Step 5.3.3.2.4

Factor $25$ out of $25 x - 125$ .

Tap for more steps...

Step 5.3.3.2.4.1

Factor $25$ out of $25 x$ .

$f (\frac{\sqrt[3]{25 (x - 5)}}{5}) = 5 (\frac{25 (x) - 125}{5^{3}}) + 5$

Step 5.3.3.2.4.2

Factor $25$ out of $- 125$ .

$f (\frac{\sqrt[3]{25 (x - 5)}}{5}) = 5 (\frac{25 x + 25 \cdot - 5}{5^{3}}) + 5$

Step 5.3.3.2.4.3

Factor $25$ out of $25 x + 25 \cdot - 5$ .

$f (\frac{\sqrt[3]{25 (x - 5)}}{5}) = 5 (\frac{25 (x - 5)}{5^{3}}) + 5$

Step 5.3.3.3

Raise $5$ to the power of $3$ .

$f (\frac{\sqrt[3]{25 (x - 5)}}{5}) = 5 (\frac{25 (x - 5)}{125}) + 5$

Step 5.3.3.4

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.3.3.4.1

Factor $5$ out of $125$ .

$f (\frac{\sqrt[3]{25 (x - 5)}}{5}) = 5 (\frac{25 (x - 5)}{5 (25)}) + 5$

Step 5.3.3.4.2

Cancel the common factor.

$f (\frac{\sqrt[3]{25 (x - 5)}}{5}) = 5 (\frac{25 (x - 5)}{5 \cdot 25}) + 5$

Step 5.3.3.4.3

Rewrite the expression.

$f (\frac{\sqrt[3]{25 (x - 5)}}{5}) = \frac{25 (x - 5)}{25} + 5$

Step 5.3.3.5

Cancel the common factor of $25$ .

Tap for more steps...

Step 5.3.3.5.1

Cancel the common factor.

$f (\frac{\sqrt[3]{25 (x - 5)}}{5}) = \frac{25 (x - 5)}{25} + 5$

Step 5.3.3.5.2

Divide $x - 5$ by $1$ .

$f (\frac{\sqrt[3]{25 (x - 5)}}{5}) = x - 5 + 5$

Step 5.3.4

Combine the opposite terms in $x - 5 + 5$ .

Tap for more steps...

Step 5.3.4.1

Add $- 5$ and $5$ .

$f (\frac{\sqrt[3]{25 (x - 5)}}{5}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (\frac{\sqrt[3]{25 (x - 5)}}{5}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\sqrt[3]{25 (x - 5)}}{5}$ is the inverse of $f (x) = 5 x^{3} + 5$ .

$f^{- 1} (x) = \frac{\sqrt[3]{25 (x - 5)}}{5}$