Find the Inverse f(x)=81/(x^4)

$f (x) = \frac{81}{x^{4}}$

Step 1

Write $f (x) = \frac{81}{x^{4}}$ as an equation.

$y = \frac{81}{x^{4}}$

Step 2

Interchange the variables.

$x = \frac{81}{y^{4}}$

Step 3

Solve for $y$ .

Rewrite the equation as $\frac{81}{y^{4}} = x$ .

$\frac{81}{y^{4}} = x$

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.

$y^{4}, 1$

The LCM of one and any expression is the expression.

$y^{4}$

Multiply each term in $\frac{81}{y^{4}} = x$ by $y^{4}$ to eliminate the fractions.

Multiply each term in $\frac{81}{y^{4}} = x$ by $y^{4}$ .

$\frac{81}{y^{4}} y^{4} = x y^{4}$

Simplify the left side.

Cancel the common factor of $y^{4}$ .

Cancel the common factor.

$\frac{81}{y^{4}} y^{4} = x y^{4}$

Rewrite the expression.

$81 = x y^{4}$

Solve the equation.

Rewrite the equation as $x y^{4} = 81$ .

$x y^{4} = 81$

Divide each term in $x y^{4} = 81$ by $x$ and simplify.

Divide each term in $x y^{4} = 81$ by $x$ .

$\frac{x y^{4}}{x} = \frac{81}{x}$

Simplify the left side.

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{x y^{4}}{x} = \frac{81}{x}$

Divide $y^{4}$ by $1$ .

$y^{4} = \frac{81}{x}$

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

$y = \pm \sqrt[4]{\frac{81}{x}}$

Simplify $\pm \sqrt[4]{\frac{81}{x}}$ .

Rewrite $\sqrt[4]{\frac{81}{x}}$ as $\frac{\sqrt[4]{81}}{\sqrt[4]{x}}$ .

$y = \pm \frac{\sqrt[4]{81}}{\sqrt[4]{x}}$

Simplify the numerator.

Rewrite $81$ as $3^{4}$ .

$y = \pm \frac{\sqrt[4]{3^{4}}}{\sqrt[4]{x}}$

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

$y = \pm \frac{3}{\sqrt[4]{x}}$

Multiply $\frac{3}{\sqrt[4]{x}}$ by $\frac{{\sqrt[4]{x}}^{3}}{{\sqrt[4]{x}}^{3}}$ .

$y = \pm \frac{3}{\sqrt[4]{x}} \cdot \frac{{\sqrt[4]{x}}^{3}}{{\sqrt[4]{x}}^{3}}$

Combine and simplify the denominator.

Multiply $\frac{3}{\sqrt[4]{x}}$ by $\frac{{\sqrt[4]{x}}^{3}}{{\sqrt[4]{x}}^{3}}$ .

$y = \pm \frac{3 {\sqrt[4]{x}}^{3}}{\sqrt[4]{x} {\sqrt[4]{x}}^{3}}$

Raise $\sqrt[4]{x}$ to the power of $1$ .

$y = \pm \frac{3 {\sqrt[4]{x}}^{3}}{{\sqrt[4]{x}}^{1} {\sqrt[4]{x}}^{3}}$

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

$y = \pm \frac{3 {\sqrt[4]{x}}^{3}}{{\sqrt[4]{x}}^{1 + 3}}$

Add $1$ and $3$ .

$y = \pm \frac{3 {\sqrt[4]{x}}^{3}}{{\sqrt[4]{x}}^{4}}$

Rewrite ${\sqrt[4]{x}}^{4}$ as $x$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

$y = \pm \frac{3 {\sqrt[4]{x}}^{3}}{{(x^{\frac{1}{4}})}^{4}}$

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

$y = \pm \frac{3 {\sqrt[4]{x}}^{3}}{x^{\frac{1}{4} \cdot 4}}$

Combine $\frac{1}{4}$ and $4$ .

$y = \pm \frac{3 {\sqrt[4]{x}}^{3}}{x^{\frac{4}{4}}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$y = \pm \frac{3 {\sqrt[4]{x}}^{3}}{x^{\frac{4}{4}}}$

Rewrite the expression.

$y = \pm \frac{3 {\sqrt[4]{x}}^{3}}{x^{1}}$

Simplify.

$y = \pm \frac{3 {\sqrt[4]{x}}^{3}}{x}$

Rewrite ${\sqrt[4]{x}}^{3}$ as $\sqrt[4]{x^{3}}$ .

$y = \pm \frac{3 \sqrt[4]{x^{3}}}{x}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$y = \frac{3 \sqrt[4]{x^{3}}}{x}$

Next, use the negative value of the $\pm$ to find the second solution.