Find the Inverse f(x)=(3x)^(-5/2)

$f (x) = {(3 x)}^{- \frac{5}{2}}$

Write $f (x) = {(3 x)}^{- \frac{5}{2}}$ as an equation.

$y = {(3 x)}^{- \frac{5}{2}}$

Interchange the variables.

$x = {(3 y)}^{- \frac{5}{2}}$

Solve for $y$ .

Rewrite the equation as ${(3 y)}^{- \frac{5}{2}} = x$ .

${(3 y)}^{- \frac{5}{2}} = x$

Raise each side of the equation to the power of $- \frac{2}{5}$ to eliminate the fractional exponent on the left side.

${({(3 y)}^{- \frac{5}{2}})}^{- \frac{2}{5}} = x^{- \frac{2}{5}}$

Simplify the exponent.

Simplify the left side.

Simplify ${({(3 y)}^{- \frac{5}{2}})}^{- \frac{2}{5}}$ .

Multiply the exponents in ${({(3 y)}^{- \frac{5}{2}})}^{- \frac{2}{5}}$ .

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

${(3 y)}^{- \frac{5}{2} (- \frac{2}{5})} = x^{- \frac{2}{5}}$

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{5}{2}$ into the numerator.

${(3 y)}^{\frac{- 5}{2} (- \frac{2}{5})} = x^{- \frac{2}{5}}$

Move the leading negative in $- \frac{2}{5}$ into the numerator.

${(3 y)}^{\frac{- 5}{2} \cdot \frac{- 2}{5}} = x^{- \frac{2}{5}}$

Factor $5$ out of $- 5$ .

${(3 y)}^{\frac{5 (- 1)}{2} \cdot \frac{- 2}{5}} = x^{- \frac{2}{5}}$

Cancel the common factor.

${(3 y)}^{\frac{5 \cdot - 1}{2} \cdot \frac{- 2}{5}} = x^{- \frac{2}{5}}$

Rewrite the expression.

${(3 y)}^{\frac{- 1}{2} \cdot - 2} = x^{- \frac{2}{5}}$

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

${(3 y)}^{\frac{- 1}{2} \cdot (2 (- 1))} = x^{- \frac{2}{5}}$

Cancel the common factor.

${(3 y)}^{\frac{- 1}{2} \cdot (2 \cdot - 1)} = x^{- \frac{2}{5}}$

Rewrite the expression.

${(3 y)}^{- - 1} = x^{- \frac{2}{5}}$

Multiply $- 1$ by $- 1$ .

${(3 y)}^{1} = x^{- \frac{2}{5}}$

Simplify.

$3 y = x^{- \frac{2}{5}}$

Simplify the right side.

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$3 y = \frac{1}{x^{\frac{2}{5}}}$

Divide each term in $3 y = \frac{1}{x^{\frac{2}{5}}}$ by $3$ and simplify.

Divide each term in $3 y = \frac{1}{x^{\frac{2}{5}}}$ by $3$ .

$\frac{3 y}{3} = \frac{\frac{1}{x^{\frac{2}{5}}}}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 y}{3} = \frac{\frac{1}{x^{\frac{2}{5}}}}{3}$

Divide $y$ by $1$ .

$y = \frac{\frac{1}{x^{\frac{2}{5}}}}{3}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{1}{x^{\frac{2}{5}}} \cdot \frac{1}{3}$

Combine.

$y = \frac{1 \cdot 1}{x^{\frac{2}{5}} \cdot 3}$

Simplify the expression.

Multiply $1$ by $1$ .

$y = \frac{1}{x^{\frac{2}{5}} \cdot 3}$

Move $3$ to the left of $x^{\frac{2}{5}}$ .

$y = \frac{1}{3 x^{\frac{2}{5}}}$

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

$f^{- 1} (x) = \frac{1}{3 x^{\frac{2}{5}}}$

Verify if $f^{- 1} (x) = \frac{1}{3 x^{\frac{2}{5}}}$ is the inverse of $f (x) = {(3 x)}^{- \frac{5}{2}}$ .

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 $f^{- 1} ({(3 x)}^{- \frac{5}{2}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} ({(3 x)}^{- \frac{5}{2}}) = \frac{1}{3 {({(3 x)}^{- \frac{5}{2}})}^{\frac{2}{5}}}$

Simplify the denominator.

Multiply the exponents in ${({(3 x)}^{- \frac{5}{2}})}^{\frac{2}{5}}$ .

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

$f^{- 1} ({(3 x)}^{- \frac{5}{2}}) = \frac{1}{3 {(3 x)}^{- \frac{5}{2} \cdot \frac{2}{5}}}$

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{5}{2}$ into the numerator.

$f^{- 1} ({(3 x)}^{- \frac{5}{2}}) = \frac{1}{3 {(3 x)}^{\frac{- 5}{2} \cdot \frac{2}{5}}}$

Factor $5$ out of $- 5$ .

$f^{- 1} ({(3 x)}^{- \frac{5}{2}}) = \frac{1}{3 {(3 x)}^{\frac{5 (- 1)}{2} \cdot \frac{2}{5}}}$

Cancel the common factor.

$f^{- 1} ({(3 x)}^{- \frac{5}{2}}) = \frac{1}{3 {(3 x)}^{\frac{5 \cdot - 1}{2} \cdot \frac{2}{5}}}$

Rewrite the expression.

$f^{- 1} ({(3 x)}^{- \frac{5}{2}}) = \frac{1}{3 {(3 x)}^{\frac{- 1}{2} \cdot 2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} ({(3 x)}^{- \frac{5}{2}}) = \frac{1}{3 {(3 x)}^{\frac{- 1}{2} \cdot 2}}$

Rewrite the expression.

$f^{- 1} ({(3 x)}^{- \frac{5}{2}}) = \frac{1}{3 {(3 x)}^{- 1}}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f^{- 1} ({(3 x)}^{- \frac{5}{2}}) = \frac{1}{3 (\frac{1}{3 x})}$

Simplify terms.

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

$f^{- 1} ({(3 x)}^{- \frac{5}{2}}) = \frac{1}{\frac{3}{3 x}}$

Reduce the expression $\frac{3}{3 x}$ by cancelling the common factors.

Cancel the common factor.

$f^{- 1} ({(3 x)}^{- \frac{5}{2}}) = \frac{1}{\frac{3}{3 x}}$

Rewrite the expression.

$f^{- 1} ({(3 x)}^{- \frac{5}{2}}) = \frac{1}{\frac{1}{x}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} ({(3 x)}^{- \frac{5}{2}}) = 1 x$

Multiply $x$ by $1$ .

$f^{- 1} ({(3 x)}^{- \frac{5}{2}}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{1}{3 x^{\frac{2}{5}}})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{1}{3 x^{\frac{2}{5}}}) = {(3 (\frac{1}{3 x^{\frac{2}{5}}}))}^{- \frac{5}{2}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$f (\frac{1}{3 x^{\frac{2}{5}}}) = {(3 (\frac{1}{3 x^{\frac{2}{5}}}))}^{- \frac{5}{2}}$

Rewrite the expression.

$f (\frac{1}{3 x^{\frac{2}{5}}}) = {(\frac{1}{x^{\frac{2}{5}}})}^{- \frac{5}{2}}$

Change the sign of the exponent by rewriting the base as its reciprocal.

$f (\frac{1}{3 x^{\frac{2}{5}}}) = {(x^{\frac{2}{5}})}^{\frac{5}{2}}$

Multiply the exponents in ${(x^{\frac{2}{5}})}^{\frac{5}{2}}$ .

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

$f (\frac{1}{3 x^{\frac{2}{5}}}) = x^{\frac{2}{5} \cdot \frac{5}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (\frac{1}{3 x^{\frac{2}{5}}}) = x^{\frac{2}{5} \cdot \frac{5}{2}}$

Rewrite the expression.

$f (\frac{1}{3 x^{\frac{2}{5}}}) = x^{\frac{1}{5} \cdot 5}$

Cancel the common factor of $5$ .

Cancel the common factor.

$f (\frac{1}{3 x^{\frac{2}{5}}}) = x^{\frac{1}{5} \cdot 5}$

Rewrite the expression.

$f (\frac{1}{3 x^{\frac{2}{5}}}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{1}{3 x^{\frac{2}{5}}}$ is the inverse of $f (x) = {(3 x)}^{- \frac{5}{2}}$ .

$f^{- 1} (x) = \frac{1}{3 x^{\frac{2}{5}}}$

Find the Inverse f(x)=(3x)^(-5/2)

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