Find the Inverse f(x)=2x^(-2/3)

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

Write $f (x) = 2 x^{- \frac{2}{3}}$ as an equation.

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

Interchange the variables.

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

Solve for $y$ .

Rewrite the equation as $2 y^{- \frac{2}{3}} = x$ .

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

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

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

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

Simplify the left side.

Move $y^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{y^{\frac{2}{3}}} = \frac{x}{2}$

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.

$y^{\frac{2}{3}}, 2$

Since $y^{\frac{2}{3}}, 2$ contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 2$ then find LCM for the variable part $y^{\frac{2}{3}}$ .

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Since $2$ has no factors besides $1$ and $2$ .

$2$ is a prime number

The LCM of $1, 2$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2$

The LCM of $y^{\frac{2}{3}}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$y^{\frac{2}{3}}$

The LCM for $y^{\frac{2}{3}}, 2$ is the numeric part $2$ multiplied by the variable part.

$2 y^{\frac{2}{3}}$

Multiply each term in $\frac{1}{y^{\frac{2}{3}}} = \frac{x}{2}$ by $2 y^{\frac{2}{3}}$ to eliminate the fractions.

Multiply each term in $\frac{1}{y^{\frac{2}{3}}} = \frac{x}{2}$ by $2 y^{\frac{2}{3}}$ .

$\frac{1}{y^{\frac{2}{3}}} (2 y^{\frac{2}{3}}) = \frac{x}{2} (2 y^{\frac{2}{3}})$

Simplify the left side.

Rewrite using the commutative property of multiplication.

$2 \frac{1}{y^{\frac{2}{3}}} y^{\frac{2}{3}} = \frac{x}{2} (2 y^{\frac{2}{3}})$

Combine $2$ and $\frac{1}{y^{\frac{2}{3}}}$ .

$\frac{2}{y^{\frac{2}{3}}} y^{\frac{2}{3}} = \frac{x}{2} (2 y^{\frac{2}{3}})$

Cancel the common factor of $y^{\frac{2}{3}}$ .

Cancel the common factor.

$\frac{2}{y^{\frac{2}{3}}} y^{\frac{2}{3}} = \frac{x}{2} (2 y^{\frac{2}{3}})$

Rewrite the expression.

$2 = \frac{x}{2} (2 y^{\frac{2}{3}})$

Simplify the right side.

Rewrite using the commutative property of multiplication.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Solve the equation.

Rewrite the equation as $x y^{\frac{2}{3}} = 2$ .

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

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

Divide each term in $x y^{\frac{2}{3}} = 2$ by $x$ .

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

Simplify the left side.

Cancel the common factor.

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

Divide $y^{\frac{2}{3}}$ by $1$ .

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

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

${(y^{\frac{2}{3}})}^{\frac{3}{2}} = \pm {(\frac{2}{x})}^{\frac{3}{2}}$

Simplify the exponent.

Simplify the left side.

Simplify ${(y^{\frac{2}{3}})}^{\frac{3}{2}}$ .

Multiply the exponents in ${(y^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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

$y^{\frac{2}{3} \cdot \frac{3}{2}} = \pm {(\frac{2}{x})}^{\frac{3}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$y^{\frac{2}{3} \cdot \frac{3}{2}} = \pm {(\frac{2}{x})}^{\frac{3}{2}}$

Rewrite the expression.

$y^{\frac{1}{3} \cdot 3} = \pm {(\frac{2}{x})}^{\frac{3}{2}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$y^{\frac{1}{3} \cdot 3} = \pm {(\frac{2}{x})}^{\frac{3}{2}}$

Rewrite the expression.

$y^{1} = \pm {(\frac{2}{x})}^{\frac{3}{2}}$

Simplify.

$y = \pm {(\frac{2}{x})}^{\frac{3}{2}}$

Simplify the right side.

Apply the product rule to $\frac{2}{x}$ .

$y = \pm \frac{2^{\frac{3}{2}}}{x^{\frac{3}{2}}}$

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.

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

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