Precalculus Examples

$f (x) = {(\frac{x^{\frac{1}{7}}}{5})}^{5}$ ; find $f^{- 1} (x)$

Step 1

Write $f (x) = {(\frac{x^{\frac{1}{7}}}{5})}^{5}$ as an equation.

$y = {(\frac{x^{\frac{1}{7}}}{5})}^{5}$

Step 2

Interchange the variables.

$x = {(\frac{y^{\frac{1}{7}}}{5})}^{5}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as ${(\frac{y^{\frac{1}{7}}}{5})}^{5} = x$ .

${(\frac{y^{\frac{1}{7}}}{5})}^{5} = x$

Step 3.2

Simplify ${(\frac{y^{\frac{1}{7}}}{5})}^{5}$ .

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Step 3.2.1

Apply the product rule to $\frac{y^{\frac{1}{7}}}{5}$ .

$\frac{{(y^{\frac{1}{7}})}^{5}}{5^{5}} = x$

Step 3.2.2

Multiply the exponents in ${(y^{\frac{1}{7}})}^{5}$ .

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Step 3.2.2.1

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

$\frac{y^{\frac{1}{7} \cdot 5}}{5^{5}} = x$

Step 3.2.2.2

Combine $\frac{1}{7}$ and $5$ .

$\frac{y^{\frac{5}{7}}}{5^{5}} = x$

Step 3.2.3

Raise $5$ to the power of $5$ .

$\frac{y^{\frac{5}{7}}}{3125} = x$

Step 3.3

Multiply both sides of the equation by $3125$ .

$3125 \frac{y^{\frac{5}{7}}}{3125} = 3125 x$

Step 3.4

Simplify the left side.

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Step 3.4.1

Cancel the common factor of $3125$ .

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Step 3.4.1.1

Cancel the common factor.

$3125 \frac{y^{\frac{5}{7}}}{3125} = 3125 x$

Step 3.4.1.2

Rewrite the expression.

$y^{\frac{5}{7}} = 3125 x$

Step 3.5

Raise each side of the equation to the power of $\frac{7}{5}$ to eliminate the fractional exponent on the left side.

${(y^{\frac{5}{7}})}^{\frac{7}{5}} = {(3125 x)}^{\frac{7}{5}}$

Step 3.6

Simplify the exponent.

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Step 3.6.1

Simplify the left side.

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Step 3.6.1.1

Simplify ${(y^{\frac{5}{7}})}^{\frac{7}{5}}$ .

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Step 3.6.1.1.1

Multiply the exponents in ${(y^{\frac{5}{7}})}^{\frac{7}{5}}$ .

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Step 3.6.1.1.1.1

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

$y^{\frac{5}{7} \cdot \frac{7}{5}} = {(3125 x)}^{\frac{7}{5}}$

Step 3.6.1.1.1.2

Cancel the common factor of $5$ .

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Step 3.6.1.1.1.2.1

Cancel the common factor.

$y^{\frac{5}{7} \cdot \frac{7}{5}} = {(3125 x)}^{\frac{7}{5}}$

Step 3.6.1.1.1.2.2

Rewrite the expression.

$y^{\frac{1}{7} \cdot 7} = {(3125 x)}^{\frac{7}{5}}$

Step 3.6.1.1.1.3

Cancel the common factor of $7$ .

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Step 3.6.1.1.1.3.1

Cancel the common factor.

$y^{\frac{1}{7} \cdot 7} = {(3125 x)}^{\frac{7}{5}}$

Step 3.6.1.1.1.3.2

Rewrite the expression.

$y^{1} = {(3125 x)}^{\frac{7}{5}}$

Step 3.6.1.1.2

Simplify.

$y = {(3125 x)}^{\frac{7}{5}}$

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

Simplify ${(3125 x)}^{\frac{7}{5}}$ .

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Step 3.6.2.1.1

Simplify the expression.

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Step 3.6.2.1.1.1

Apply the product rule to $3125 x$ .

$y = 3125^{\frac{7}{5}} x^{\frac{7}{5}}$

Step 3.6.2.1.1.2

Rewrite $3125$ as $5^{5}$ .

$y = {(5^{5})}^{\frac{7}{5}} x^{\frac{7}{5}}$

Step 3.6.2.1.1.3

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

$y = 5^{5 (\frac{7}{5})} x^{\frac{7}{5}}$

Step 3.6.2.1.2

Cancel the common factor of $5$ .

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Step 3.6.2.1.2.1

Cancel the common factor.

$y = 5^{5 (\frac{7}{5})} x^{\frac{7}{5}}$

Step 3.6.2.1.2.2

Rewrite the expression.

$y = 5^{7} x^{\frac{7}{5}}$

Step 3.6.2.1.3

Raise $5$ to the power of $7$ .

$y = 78125 x^{\frac{7}{5}}$

Step 4

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

$f^{- 1} (x) = 78125 x^{\frac{7}{5}}$

Step 5

Verify if $f^{- 1} (x) = 78125 x^{\frac{7}{5}}$ is the inverse of $f (x) = {(\frac{x^{\frac{1}{7}}}{5})}^{5}$ .

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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 $f^{- 1} ({(\frac{x^{\frac{1}{7}}}{5})}^{5})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} ({(\frac{x^{\frac{1}{7}}}{5})}^{5}) = 78125 {({(\frac{x^{\frac{1}{7}}}{5})}^{5})}^{\frac{7}{5}}$

Step 5.2.3

Apply basic rules of exponents.

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Step 5.2.3.1

Multiply the exponents in ${({(\frac{x^{\frac{1}{7}}}{5})}^{5})}^{\frac{7}{5}}$ .

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Step 5.2.3.1.1

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

$f^{- 1} ({(\frac{x^{\frac{1}{7}}}{5})}^{5}) = 78125 {(\frac{x^{\frac{1}{7}}}{5})}^{5 (\frac{7}{5})}$

Step 5.2.3.1.2

Cancel the common factor of $5$ .

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Step 5.2.3.1.2.1

Cancel the common factor.

$f^{- 1} ({(\frac{x^{\frac{1}{7}}}{5})}^{5}) = 78125 {(\frac{x^{\frac{1}{7}}}{5})}^{5 (\frac{7}{5})}$

Step 5.2.3.1.2.2

Rewrite the expression.

$f^{- 1} ({(\frac{x^{\frac{1}{7}}}{5})}^{5}) = 78125 {(\frac{x^{\frac{1}{7}}}{5})}^{7}$

Step 5.2.3.2

Apply the product rule to $\frac{x^{\frac{1}{7}}}{5}$ .

$f^{- 1} ({(\frac{x^{\frac{1}{7}}}{5})}^{5}) = 78125 (\frac{{(x^{\frac{1}{7}})}^{7}}{5^{7}})$

Step 5.2.4

Simplify the numerator.

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Step 5.2.4.1

Multiply the exponents in ${(x^{\frac{1}{7}})}^{7}$ .

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Step 5.2.4.1.1

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

$f^{- 1} ({(\frac{x^{\frac{1}{7}}}{5})}^{5}) = 78125 (\frac{x^{\frac{1}{7} \cdot 7}}{5^{7}})$

Step 5.2.4.1.2

Cancel the common factor of $7$ .

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Step 5.2.4.1.2.1

Cancel the common factor.

$f^{- 1} ({(\frac{x^{\frac{1}{7}}}{5})}^{5}) = 78125 (\frac{x^{\frac{1}{7} \cdot 7}}{5^{7}})$

Step 5.2.4.1.2.2

Rewrite the expression.

$f^{- 1} ({(\frac{x^{\frac{1}{7}}}{5})}^{5}) = 78125 (\frac{x}{5^{7}})$

Step 5.2.4.2

Simplify.

$f^{- 1} ({(\frac{x^{\frac{1}{7}}}{5})}^{5}) = 78125 (\frac{x}{5^{7}})$

Step 5.2.5

Reduce the expression by cancelling the common factors.

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Step 5.2.5.1

Raise $5$ to the power of $7$ .

$f^{- 1} ({(\frac{x^{\frac{1}{7}}}{5})}^{5}) = 78125 (\frac{x}{78125})$

Step 5.2.5.2

Cancel the common factor of $78125$ .

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Step 5.2.5.2.1

Cancel the common factor.

$f^{- 1} ({(\frac{x^{\frac{1}{7}}}{5})}^{5}) = 78125 (\frac{x}{78125})$

Step 5.2.5.2.2

Rewrite the expression.

$f^{- 1} ({(\frac{x^{\frac{1}{7}}}{5})}^{5}) = x$

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 $f (78125 x^{\frac{7}{5}})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (78125 x^{\frac{7}{5}}) = {(\frac{{(78125 x^{\frac{7}{5}})}^{\frac{1}{7}}}{5})}^{5}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Apply the product rule to $78125 x^{\frac{7}{5}}$ .

$f (78125 x^{\frac{7}{5}}) = {(\frac{78125^{\frac{1}{7}} {(x^{\frac{7}{5}})}^{\frac{1}{7}}}{5})}^{5}$

Step 5.3.3.2

Rewrite $78125$ as $5^{7}$ .

$f (78125 x^{\frac{7}{5}}) = {(\frac{{(5^{7})}^{\frac{1}{7}} {(x^{\frac{7}{5}})}^{\frac{1}{7}}}{5})}^{5}$

Step 5.3.3.3

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

$f (78125 x^{\frac{7}{5}}) = {(\frac{5^{7 (\frac{1}{7})} {(x^{\frac{7}{5}})}^{\frac{1}{7}}}{5})}^{5}$

Step 5.3.3.4

Cancel the common factor of $7$ .

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Step 5.3.3.4.1

Cancel the common factor.

$f (78125 x^{\frac{7}{5}}) = {(\frac{5^{7 (\frac{1}{7})} {(x^{\frac{7}{5}})}^{\frac{1}{7}}}{5})}^{5}$

Step 5.3.3.4.2

Rewrite the expression.

$f (78125 x^{\frac{7}{5}}) = {(\frac{5 {(x^{\frac{7}{5}})}^{\frac{1}{7}}}{5})}^{5}$

Step 5.3.3.5

Evaluate the exponent.

$f (78125 x^{\frac{7}{5}}) = {(\frac{5 {(x^{\frac{7}{5}})}^{\frac{1}{7}}}{5})}^{5}$

Step 5.3.3.6

Multiply the exponents in ${(x^{\frac{7}{5}})}^{\frac{1}{7}}$ .

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Step 5.3.3.6.1

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

$f (78125 x^{\frac{7}{5}}) = {(\frac{5 x^{\frac{7}{5} \cdot \frac{1}{7}}}{5})}^{5}$

Step 5.3.3.6.2

Cancel the common factor of $7$ .

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Step 5.3.3.6.2.1

Cancel the common factor.

$f (78125 x^{\frac{7}{5}}) = {(\frac{5 x^{\frac{7}{5} \cdot \frac{1}{7}}}{5})}^{5}$

Step 5.3.3.6.2.2

Rewrite the expression.

$f (78125 x^{\frac{7}{5}}) = {(\frac{5 x^{\frac{1}{5}}}{5})}^{5}$

Step 5.3.4

Reduce the expression by cancelling the common factors.

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Step 5.3.4.1

Cancel the common factor.

$f (78125 x^{\frac{7}{5}}) = {(\frac{5 x^{\frac{1}{5}}}{5})}^{5}$

Step 5.3.4.2

Simplify the expression.

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Step 5.3.4.2.1

Divide $x^{\frac{1}{5}}$ by $1$ .

$f (78125 x^{\frac{7}{5}}) = {(x^{\frac{1}{5}})}^{5}$

Step 5.3.4.2.2

Multiply the exponents in ${(x^{\frac{1}{5}})}^{5}$ .

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Step 5.3.4.2.2.1

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

$f (78125 x^{\frac{7}{5}}) = x^{\frac{1}{5} \cdot 5}$

Step 5.3.4.2.2.2

Cancel the common factor of $5$ .

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Step 5.3.4.2.2.2.1

Cancel the common factor.

$f (78125 x^{\frac{7}{5}}) = x^{\frac{1}{5} \cdot 5}$

Step 5.3.4.2.2.2.2

Rewrite the expression.

$f (78125 x^{\frac{7}{5}}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 78125 x^{\frac{7}{5}}$ is the inverse of $f (x) = {(\frac{x^{\frac{1}{7}}}{5})}^{5}$ .

$f^{- 1} (x) = 78125 x^{\frac{7}{5}}$