Find the Inverse f(x)=19/(x^3)

$f (x) = \frac{19}{x^{3}}$

Step 1

Write $f (x) = \frac{19}{x^{3}}$ as an equation.

$y = \frac{19}{x^{3}}$

Step 2

Interchange the variables.

$x = \frac{19}{y^{3}}$

Step 3

Solve for $y$ .

Rewrite the equation as $\frac{19}{y^{3}} = x$ .

$\frac{19}{y^{3}} = x$

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y^{3}, 1$

The LCM of one and any expression is the expression.

$y^{3}$

Multiply each term in $\frac{19}{y^{3}} = x$ by $y^{3}$ to eliminate the fractions.

Multiply each term in $\frac{19}{y^{3}} = x$ by $y^{3}$ .

$\frac{19}{y^{3}} y^{3} = x y^{3}$

Simplify the left side.

Cancel the common factor of $y^{3}$ .

Cancel the common factor.

$\frac{19}{y^{3}} y^{3} = x y^{3}$

Rewrite the expression.

$19 = x y^{3}$

Solve the equation.

Rewrite the equation as $x y^{3} = 19$ .

$x y^{3} = 19$

Divide each term in $x y^{3} = 19$ by $x$ and simplify.

Divide each term in $x y^{3} = 19$ by $x$ .

$\frac{x y^{3}}{x} = \frac{19}{x}$

Simplify the left side.

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{x y^{3}}{x} = \frac{19}{x}$

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

$y^{3} = \frac{19}{x}$

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

$y = \sqrt[3]{\frac{19}{x}}$

Simplify $\sqrt[3]{\frac{19}{x}}$ .

Rewrite $\sqrt[3]{\frac{19}{x}}$ as $\frac{\sqrt[3]{19}}{\sqrt[3]{x}}$ .

$y = \frac{\sqrt[3]{19}}{\sqrt[3]{x}}$

Multiply $\frac{\sqrt[3]{19}}{\sqrt[3]{x}}$ by $\frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$ .

$y = \frac{\sqrt[3]{19}}{\sqrt[3]{x}} \cdot \frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt[3]{19}}{\sqrt[3]{x}}$ by $\frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$ .

$y = \frac{\sqrt[3]{19} {\sqrt[3]{x}}^{2}}{\sqrt[3]{x} {\sqrt[3]{x}}^{2}}$

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

$y = \frac{\sqrt[3]{19} {\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{1} {\sqrt[3]{x}}^{2}}$

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

$y = \frac{\sqrt[3]{19} {\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{1 + 2}}$

Add $1$ and $2$ .

$y = \frac{\sqrt[3]{19} {\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{3}}$

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

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

$y = \frac{\sqrt[3]{19} {\sqrt[3]{x}}^{2}}{{(x^{\frac{1}{3}})}^{3}}$

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

$y = \frac{\sqrt[3]{19} {\sqrt[3]{x}}^{2}}{x^{\frac{1}{3} \cdot 3}}$

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

$y = \frac{\sqrt[3]{19} {\sqrt[3]{x}}^{2}}{x^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$y = \frac{\sqrt[3]{19} {\sqrt[3]{x}}^{2}}{x^{\frac{3}{3}}}$

Rewrite the expression.

$y = \frac{\sqrt[3]{19} {\sqrt[3]{x}}^{2}}{x^{1}}$

Simplify.

$y = \frac{\sqrt[3]{19} {\sqrt[3]{x}}^{2}}{x}$

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

$y = \frac{\sqrt[3]{19} \sqrt[3]{x^{2}}}{x}$

Combine using the product rule for radicals.

$y = \frac{\sqrt[3]{19 x^{2}}}{x}$

Step 4

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

$f^{- 1} (x) = \frac{\sqrt[3]{19 x^{2}}}{x}$

Step 5

Verify if $f^{- 1} (x) = \frac{\sqrt[3]{19 x^{2}}}{x}$ is the inverse of $f (x) = \frac{19}{x^{3}}$ .

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

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

Set up the composite result function.

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

Evaluate $f^{- 1} (\frac{19}{x^{3}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{19}{x^{3}}) = \frac{\sqrt[3]{19 {(\frac{19}{x^{3}})}^{2}}}{\frac{19}{x^{3}}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{19}{x^{3}}) = \sqrt[3]{19 {(\frac{19}{x^{3}})}^{2}} (\frac{x^{3}}{19})$

Simplify the expression.

Apply the product rule to $\frac{19}{x^{3}}$ .

$f^{- 1} (\frac{19}{x^{3}}) = \sqrt[3]{19 (\frac{19^{2}}{{(x^{3})}^{2}})} (\frac{x^{3}}{19})$

Raise $19$ to the power of $2$ .

$f^{- 1} (\frac{19}{x^{3}}) = \sqrt[3]{19 (\frac{361}{{(x^{3})}^{2}})} (\frac{x^{3}}{19})$

Multiply the exponents in ${(x^{3})}^{2}$ .

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

$f^{- 1} (\frac{19}{x^{3}}) = \sqrt[3]{19 (\frac{361}{x^{3 \cdot 2}})} (\frac{x^{3}}{19})$

Multiply $3$ by $2$ .

$f^{- 1} (\frac{19}{x^{3}}) = \sqrt[3]{19 (\frac{361}{x^{6}})} (\frac{x^{3}}{19})$

Combine $19$ and $\frac{361}{x^{6}}$ .

$f^{- 1} (\frac{19}{x^{3}}) = \sqrt[3]{\frac{19 \cdot 361}{x^{6}}} (\frac{x^{3}}{19})$

Simplify the expression.

Multiply $19$ by $361$ .

$f^{- 1} (\frac{19}{x^{3}}) = \sqrt[3]{\frac{6859}{x^{6}}} (\frac{x^{3}}{19})$

Rewrite $6859$ as $19^{3}$ .

$f^{- 1} (\frac{19}{x^{3}}) = \sqrt[3]{\frac{19^{3}}{x^{6}}} (\frac{x^{3}}{19})$

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

$f^{- 1} (\frac{19}{x^{3}}) = \sqrt[3]{\frac{19^{3}}{{(x^{2})}^{3}}} (\frac{x^{3}}{19})$

Rewrite $\frac{19^{3}}{{(x^{2})}^{3}}$ as ${(\frac{19}{x^{2}})}^{3}$ .

$f^{- 1} (\frac{19}{x^{3}}) = \sqrt[3]{{(\frac{19}{x^{2}})}^{3}} (\frac{x^{3}}{19})$

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

$f^{- 1} (\frac{19}{x^{3}}) = \frac{19}{x^{2}} \cdot \frac{x^{3}}{19}$

Combine.

$f^{- 1} (\frac{19}{x^{3}}) = \frac{19 x^{3}}{x^{2} \cdot 19}$

Cancel the common factor of $19$ .

Cancel the common factor.

$f^{- 1} (\frac{19}{x^{3}}) = \frac{19 x^{3}}{x^{2} \cdot 19}$

Rewrite the expression.

$f^{- 1} (\frac{19}{x^{3}}) = \frac{x^{3}}{x^{2}}$

Cancel the common factor of $x^{3}$ and $x^{2}$ .

Factor $x^{2}$ out of $x^{3}$ .

$f^{- 1} (\frac{19}{x^{3}}) = \frac{x^{2} x}{x^{2}}$

Cancel the common factors.

Multiply by $1$ .

$f^{- 1} (\frac{19}{x^{3}}) = \frac{x^{2} x}{x^{2} \cdot 1}$

Cancel the common factor.

$f^{- 1} (\frac{19}{x^{3}}) = \frac{x^{2} x}{x^{2} \cdot 1}$

Rewrite the expression.

$f^{- 1} (\frac{19}{x^{3}}) = \frac{x}{1}$

Divide $x$ by $1$ .

$f^{- 1} (\frac{19}{x^{3}}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{\sqrt[3]{19 x^{2}}}{x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\sqrt[3]{19 x^{2}}}{x}) = \frac{19}{{(\frac{\sqrt[3]{19 x^{2}}}{x})}^{3}}$

Simplify the denominator.

Apply the product rule to $\frac{\sqrt[3]{19 x^{2}}}{x}$ .

$f (\frac{\sqrt[3]{19 x^{2}}}{x}) = \frac{19}{\frac{{\sqrt[3]{19 x^{2}}}^{3}}{x^{3}}}$

Rewrite ${\sqrt[3]{19 x^{2}}}^{3}$ as $19 x^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{19 x^{2}}$ as ${(19 x^{2})}^{\frac{1}{3}}$ .

$f (\frac{\sqrt[3]{19 x^{2}}}{x}) = \frac{19}{\frac{{({(19 x^{2})}^{\frac{1}{3}})}^{3}}{x^{3}}}$

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

$f (\frac{\sqrt[3]{19 x^{2}}}{x}) = \frac{19}{\frac{{(19 x^{2})}^{\frac{1}{3} \cdot 3}}{x^{3}}}$

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

$f (\frac{\sqrt[3]{19 x^{2}}}{x}) = \frac{19}{\frac{{(19 x^{2})}^{\frac{3}{3}}}{x^{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$f (\frac{\sqrt[3]{19 x^{2}}}{x}) = \frac{19}{\frac{{(19 x^{2})}^{\frac{3}{3}}}{x^{3}}}$

Rewrite the expression.

$f (\frac{\sqrt[3]{19 x^{2}}}{x}) = \frac{19}{\frac{19 x^{2}}{x^{3}}}$

Simplify.

$f (\frac{\sqrt[3]{19 x^{2}}}{x}) = \frac{19}{\frac{19 x^{2}}{x^{3}}}$

Cancel the common factor of $x^{2}$ and $x^{3}$ .

Factor $x^{2}$ out of $19 x^{2}$ .

$f (\frac{\sqrt[3]{19 x^{2}}}{x}) = \frac{19}{\frac{x^{2} \cdot 19}{x^{3}}}$

Cancel the common factors.

Factor $x^{2}$ out of $x^{3}$ .

$f (\frac{\sqrt[3]{19 x^{2}}}{x}) = \frac{19}{\frac{x^{2} \cdot 19}{x^{2} x}}$

Cancel the common factor.

$f (\frac{\sqrt[3]{19 x^{2}}}{x}) = \frac{19}{\frac{x^{2} \cdot 19}{x^{2} x}}$

Rewrite the expression.

$f (\frac{\sqrt[3]{19 x^{2}}}{x}) = \frac{19}{\frac{19}{x}}$

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{\sqrt[3]{19 x^{2}}}{x}) = 19 (\frac{x}{19})$

Cancel the common factor of $19$ .

Cancel the common factor.

$f (\frac{\sqrt[3]{19 x^{2}}}{x}) = 19 (\frac{x}{19})$

Rewrite the expression.

$f (\frac{\sqrt[3]{19 x^{2}}}{x}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\sqrt[3]{19 x^{2}}}{x}$ is the inverse of $f (x) = \frac{19}{x^{3}}$ .

$f^{- 1} (x) = \frac{\sqrt[3]{19 x^{2}}}{x}$

Find the Inverse f(x)=19/(x^3)

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