Find the Inverse y=3 square root of x

$y = 3 \sqrt{x}$

Interchange the variables.

$x = 3 \sqrt{y}$

Solve for $y$ .

Rewrite the equation as $3 \sqrt{y} = x$ .

$3 \sqrt{y} = x$

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

Divide each term in $3 \sqrt{y} = x$ by $3$ .

$\frac{3 \sqrt{y}}{3} = \frac{x}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 \sqrt{y}}{3} = \frac{x}{3}$

Divide $\sqrt{y}$ by $1$ .

$\sqrt{y} = \frac{x}{3}$

To remove the radical on the left side of the equation, square both sides of the equation.

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

Simplify each side of the equation.

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

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

Simplify the left side.

Simplify ${(y^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${(y^{\frac{1}{2}})}^{2}$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

Raise $3$ to the power of $2$ .

$y = \frac{x^{2}}{9}$

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

$f^{- 1} (x) = \frac{x^{2}}{9}$

Verify if $f^{- 1} (x) = \frac{x^{2}}{9}$ is the inverse of $f (x) = 3 \sqrt{x}$ .

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

$f^{- 1} (3 \sqrt{x}) = \frac{{(3 \sqrt{x})}^{2}}{9}$

Simplify the numerator.

Apply the product rule to $3 \sqrt{x}$ .

$f^{- 1} (3 \sqrt{x}) = \frac{3^{2} {\sqrt{x}}^{2}}{9}$

Raise $3$ to the power of $2$ .

$f^{- 1} (3 \sqrt{x}) = \frac{9 {\sqrt{x}}^{2}}{9}$

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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

$f^{- 1} (3 \sqrt{x}) = \frac{9 {(x^{\frac{1}{2}})}^{2}}{9}$

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

$f^{- 1} (3 \sqrt{x}) = \frac{9 x^{\frac{1}{2} \cdot 2}}{9}$

Combine $\frac{1}{2}$ and $2$ .

$f^{- 1} (3 \sqrt{x}) = \frac{9 x^{\frac{2}{2}}}{9}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (3 \sqrt{x}) = \frac{9 x^{\frac{2}{2}}}{9}$

Rewrite the expression.

$f^{- 1} (3 \sqrt{x}) = \frac{9 x}{9}$

Simplify.

$f^{- 1} (3 \sqrt{x}) = \frac{9 x}{9}$

Cancel the common factor of $9$ .

Cancel the common factor.

$f^{- 1} (3 \sqrt{x}) = \frac{9 x}{9}$

Divide $x$ by $1$ .

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

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

Set up the composite result function.

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

Evaluate $f (\frac{x^{2}}{9})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{2}}{9}) = 3 \sqrt{\frac{x^{2}}{9}}$

Remove parentheses.

$f (\frac{x^{2}}{9}) = 3 \sqrt{\frac{x^{2}}{9}}$

Rewrite $9$ as $3^{2}$ .

$f (\frac{x^{2}}{9}) = 3 \sqrt{\frac{x^{2}}{3^{2}}}$

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

$f (\frac{x^{2}}{9}) = 3 \sqrt{{(\frac{x}{3})}^{2}}$

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

$f (\frac{x^{2}}{9}) = 3 (\frac{x}{3})$

Cancel the common factor of $3$ .

Cancel the common factor.

$f (\frac{x^{2}}{9}) = 3 (\frac{x}{3})$

Rewrite the expression.

$f (\frac{x^{2}}{9}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{2}}{9}$ is the inverse of $f (x) = 3 \sqrt{x}$ .

$f^{- 1} (x) = \frac{x^{2}}{9}$

Find the Inverse y=3 square root of x

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