Precalculus Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

Raise each side of the equation to the power of $\frac{5}{3}$ to eliminate the fractional exponent on the left side.

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

Step 3.3

Simplify the left side.

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Step 3.3.1

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

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Step 3.3.1.1

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

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Step 3.3.1.1.1

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

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

Step 3.3.1.1.2

Cancel the common factor of $3$ .

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Step 3.3.1.1.2.1

Cancel the common factor.

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

Step 3.3.1.1.2.2

Rewrite the expression.

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

Step 3.3.1.1.3

Cancel the common factor of $5$ .

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Step 3.3.1.1.3.1

Cancel the common factor.

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

Step 3.3.1.1.3.2

Rewrite the expression.

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

Step 3.3.1.2

Simplify.

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

Step 4

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

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

Step 5

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

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Step 5.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

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

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

Step 5.2.3

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

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Step 5.2.3.1

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

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

Step 5.2.3.2

Cancel the common factor of $3$ .

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Step 5.2.3.2.1

Cancel the common factor.

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

Step 5.2.3.2.2

Rewrite the expression.

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

Step 5.2.3.3

Cancel the common factor of $5$ .

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Step 5.2.3.3.1

Cancel the common factor.

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

Step 5.2.3.3.2

Rewrite the expression.

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

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

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

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

Step 5.3.3

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

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Step 5.3.3.1

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

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

Step 5.3.3.2

Cancel the common factor of $5$ .

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Step 5.3.3.2.1

Cancel the common factor.

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

Step 5.3.3.2.2

Rewrite the expression.

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

Step 5.3.3.3

Cancel the common factor of $3$ .

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Step 5.3.3.3.1

Cancel the common factor.

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

Step 5.3.3.3.2

Rewrite the expression.

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

Step 5.4

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

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