Precalculus Examples

$f (x) = {(2 x + 5)}^{3}$

Step 1

Write $f (x) = {(2 x + 5)}^{3}$ as an equation.

$y = {(2 x + 5)}^{3}$

Step 2

Interchange the variables.

$x = {(2 y + 5)}^{3}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as ${(2 y + 5)}^{3} = x$ .

${(2 y + 5)}^{3} = x$

Step 3.2

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

$2 y + 5 = \sqrt[3]{x}$

Step 3.3

Subtract $5$ from both sides of the equation.

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

Step 3.4

Divide each term in $2 y = \sqrt[3]{x} - 5$ by $2$ and simplify.

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Step 3.4.1

Divide each term in $2 y = \sqrt[3]{x} - 5$ by $2$ .

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

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.1.1

Cancel the common factor.

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

Step 3.4.2.1.2

Divide $y$ by $1$ .

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

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = \frac{\sqrt[3]{x}}{2} - \frac{5}{2}$ is the inverse of $f (x) = {(2 x + 5)}^{3}$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} ({(2 x + 5)}^{3})$ by substituting in the value of $f$ into $f^{- 1}$ .

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

Step 5.2.3

Combine the numerators over the common denominator.

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

Step 5.2.4

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

$f^{- 1} ({(2 x + 5)}^{3}) = \frac{2 x + 5 - 5}{2}$

Step 5.2.5

Combine the opposite terms in $2 x + 5 - 5$ .

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Step 5.2.5.1

Subtract $5$ from $5$ .

$f^{- 1} ({(2 x + 5)}^{3}) = \frac{2 x + 0}{2}$

Step 5.2.5.2

Add $2 x$ and $0$ .

$f^{- 1} ({(2 x + 5)}^{3}) = \frac{2 x}{2}$

Step 5.2.6

Cancel the common factor of $2$ .

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Step 5.2.6.1

Cancel the common factor.

$f^{- 1} ({(2 x + 5)}^{3}) = \frac{2 x}{2}$

Step 5.2.6.2

Divide $x$ by $1$ .

$f^{- 1} ({(2 x + 5)}^{3}) = x$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (\frac{\sqrt[3]{x}}{2} - \frac{5}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

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

Step 5.3.3.2

Cancel the common factor of $2$ .

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Step 5.3.3.2.1

Cancel the common factor.

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

Step 5.3.3.2.2

Rewrite the expression.

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

Step 5.3.3.3

Cancel the common factor of $2$ .

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Step 5.3.3.3.1

Move the leading negative in $- \frac{5}{2}$ into the numerator.

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

Step 5.3.3.3.2

Cancel the common factor.

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

Step 5.3.3.3.3

Rewrite the expression.

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

Step 5.3.4

Simplify terms.

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Step 5.3.4.1

Combine the opposite terms in $\sqrt[3]{x} - 5 + 5$ .

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Step 5.3.4.1.1

Add $- 5$ and $5$ .

$f (\frac{\sqrt[3]{x}}{2} - \frac{5}{2}) = {(\sqrt[3]{x} + 0)}^{3}$

Step 5.3.4.1.2

Add $\sqrt[3]{x}$ and $0$ .

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

Step 5.3.4.2

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

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Step 5.3.4.2.1

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

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

Step 5.3.4.2.2

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

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

Step 5.3.4.2.3

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

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

Step 5.3.4.2.4

Cancel the common factor of $3$ .

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Step 5.3.4.2.4.1

Cancel the common factor.

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

Step 5.3.4.2.4.2

Rewrite the expression.

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

Step 5.3.4.2.5

Simplify.

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

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\sqrt[3]{x}}{2} - \frac{5}{2}$ is the inverse of $f (x) = {(2 x + 5)}^{3}$ .

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