Find the Inverse f(x)=(4x)^(2/3)

$f (x) = {(4 x)}^{\frac{2}{3}}$

Write $f (x) = {(4 x)}^{\frac{2}{3}}$ as an equation.

$y = {(4 x)}^{\frac{2}{3}}$

Interchange the variables.

$x = {(4 y)}^{\frac{2}{3}}$

Solve for $y$ .

Rewrite the equation as ${(4 y)}^{\frac{2}{3}} = x$ .

${(4 y)}^{\frac{2}{3}} = x$

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

${({(4 y)}^{\frac{2}{3}})}^{\frac{3}{2}} = \pm x^{\frac{3}{2}}$

Simplify the left side.

Simplify ${({(4 y)}^{\frac{2}{3}})}^{\frac{3}{2}}$ .

Multiply the exponents in ${({(4 y)}^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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

${(4 y)}^{\frac{2}{3} \cdot \frac{3}{2}} = \pm x^{\frac{3}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(4 y)}^{\frac{2}{3} \cdot \frac{3}{2}} = \pm x^{\frac{3}{2}}$

Rewrite the expression.

${(4 y)}^{\frac{1}{3} \cdot 3} = \pm x^{\frac{3}{2}}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(4 y)}^{\frac{1}{3} \cdot 3} = \pm x^{\frac{3}{2}}$

Rewrite the expression.

${(4 y)}^{1} = \pm x^{\frac{3}{2}}$

Simplify.

$4 y = \pm x^{\frac{3}{2}}$

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.

$4 y = x^{\frac{3}{2}}$

Divide each term in $4 y = x^{\frac{3}{2}}$ by $4$ and simplify.

Divide each term in $4 y = x^{\frac{3}{2}}$ by $4$ .

$\frac{4 y}{4} = \frac{x^{\frac{3}{2}}}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 y}{4} = \frac{x^{\frac{3}{2}}}{4}$

Divide $y$ by $1$ .

$y = \frac{x^{\frac{3}{2}}}{4}$

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

$4 y = - x^{\frac{3}{2}}$

Divide each term in $4 y = - x^{\frac{3}{2}}$ by $4$ and simplify.

Divide each term in $4 y = - x^{\frac{3}{2}}$ by $4$ .

$\frac{4 y}{4} = \frac{- x^{\frac{3}{2}}}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 y}{4} = \frac{- x^{\frac{3}{2}}}{4}$

Divide $y$ by $1$ .

$y = \frac{- x^{\frac{3}{2}}}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$y = - \frac{x^{\frac{3}{2}}}{4}$

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

$y = \frac{x^{\frac{3}{2}}}{4}$

$y = - \frac{x^{\frac{3}{2}}}{4}$

$y = \frac{x^{\frac{3}{2}}}{4}$

$y = - \frac{x^{\frac{3}{2}}}{4}$

$y = \frac{x^{\frac{3}{2}}}{4}$

$y = - \frac{x^{\frac{3}{2}}}{4}$

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

$f^{- 1} (x) = \frac{x^{\frac{3}{2}}}{4}, - \frac{x^{\frac{3}{2}}}{4}$

Verify if $f^{- 1} (x) = \frac{x^{\frac{3}{2}}}{4}, - \frac{x^{\frac{3}{2}}}{4}$ is the inverse of $f (x) = {(4 x)}^{\frac{2}{3}}$ .

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = {(4 x)}^{\frac{2}{3}}$ and $f^{- 1} (x) = \frac{x^{\frac{3}{2}}}{4}, - \frac{x^{\frac{3}{2}}}{4}$ and compare them.

Find the range of $f (x) = {(4 x)}^{\frac{2}{3}}$ .

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[0, \infty)$

Find the domain of $\frac{x^{\frac{3}{2}}}{4}$ .

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{\sqrt{x^{3}}}{4}$

Set the radicand in $\sqrt{x^{3}}$ greater than or equal to $0$ to find where the expression is defined.

$x^{3} \geq 0$

Solve for $x$ .

Take the cube root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt[3]{x^{3}} \geq \sqrt[3]{0}$

Simplify the equation.

Simplify the left side.

Pull terms out from under the radical.

$x \geq \sqrt[3]{0}$

Simplify the right side.

Simplify $\sqrt[3]{0}$ .

Rewrite $0$ as $0^{3}$ .

$x \geq \sqrt[3]{0^{3}}$

Pull terms out from under the radical.

$x \geq 0$

The domain is all values of $x$ that make the expression defined.

$[0, \infty)$

Find the domain of $f (x) = {(4 x)}^{\frac{2}{3}}$ .

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\sqrt[3]{{(4 x)}^{2}}$

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

$(- \infty, \infty)$

Find the range of $\frac{x^{\frac{3}{2}}}{4}$ .

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, \infty)$

Since the range of $f^{- 1} (x) = \frac{x^{\frac{3}{2}}}{4}, - \frac{x^{\frac{3}{2}}}{4}$ is not equal to the domain of $f (x) = {(4 x)}^{\frac{2}{3}}$ , then $f^{- 1} (x) = \frac{x^{\frac{3}{2}}}{4}, - \frac{x^{\frac{3}{2}}}{4}$ is not an inverse of $f (x) = {(4 x)}^{\frac{2}{3}}$ .

There is no inverse

Find the Inverse f(x)=(4x)^(2/3)

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