Find the Inverse f(x)=(x^(1/3))/5+7

$f (x) = \frac{x^{\frac{1}{3}}}{5} + 7$

Write $f (x) = \frac{x^{\frac{1}{3}}}{5} + 7$ as an equation.

$y = \frac{x^{\frac{1}{3}}}{5} + 7$

Interchange the variables.

$x = \frac{y^{\frac{1}{3}}}{5} + 7$

Solve for $y$ .

Rewrite the equation as $\frac{y^{\frac{1}{3}}}{5} + 7 = x$ .

$\frac{y^{\frac{1}{3}}}{5} + 7 = x$

Move all terms not containing $y$ to the right side of the equation.

Subtract $7$ from both sides of the equation.

$\frac{y^{\frac{1}{3}}}{5} - x = - 7$

Add $x$ to both sides of the equation.

$\frac{y^{\frac{1}{3}}}{5} = - 7 + x$

Multiply both sides of the equation by $5$ .

$5 \frac{y^{\frac{1}{3}}}{5} = 5 (- 7 + x)$

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$5 \frac{y^{\frac{1}{3}}}{5} = 5 (- 7 + x)$

Rewrite the expression.

$y^{\frac{1}{3}} = 5 (- 7 + x)$

Simplify the right side.

Simplify $5 (- 7 + x)$ .

Apply the distributive property.

$y^{\frac{1}{3}} = 5 \cdot - 7 + 5 x$

Multiply $5$ by $- 7$ .

$y^{\frac{1}{3}} = - 35 + 5 x$

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${(y^{\frac{1}{3}})}^{3} = {(- 35 + 5 x)}^{3}$

Simplify the exponent.

Simplify the left side.

Simplify ${(y^{\frac{1}{3}})}^{3}$ .

Multiply the exponents in ${(y^{\frac{1}{3}})}^{3}$ .

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

$y^{\frac{1}{3} \cdot 3} = {(- 35 + 5 x)}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$y^{\frac{1}{3} \cdot 3} = {(- 35 + 5 x)}^{3}$

Rewrite the expression.

$y^{1} = {(- 35 + 5 x)}^{3}$

Simplify.

$y = {(- 35 + 5 x)}^{3}$

Simplify the right side.

Simplify ${(- 35 + 5 x)}^{3}$ .

Use the Binomial Theorem.

$y = {(- 35)}^{3} + 3 {(- 35)}^{2} (5 x) + 3 \cdot - 35 {(5 x)}^{2} + {(5 x)}^{3}$

Simplify each term.

Raise $- 35$ to the power of $3$ .

$y = - 42875 + 3 {(- 35)}^{2} (5 x) + 3 \cdot - 35 {(5 x)}^{2} + {(5 x)}^{3}$

Raise $- 35$ to the power of $2$ .

$y = - 42875 + 3 \cdot 1225 (5 x) + 3 \cdot - 35 {(5 x)}^{2} + {(5 x)}^{3}$

Multiply $3$ by $1225$ .

$y = - 42875 + 3675 (5 x) + 3 \cdot - 35 {(5 x)}^{2} + {(5 x)}^{3}$

Multiply $5$ by $3675$ .

$y = - 42875 + 18375 x + 3 \cdot - 35 {(5 x)}^{2} + {(5 x)}^{3}$

Multiply $3$ by $- 35$ .

$y = - 42875 + 18375 x - 105 {(5 x)}^{2} + {(5 x)}^{3}$

Apply the product rule to $5 x$ .

$y = - 42875 + 18375 x - 105 (5^{2} x^{2}) + {(5 x)}^{3}$

Raise $5$ to the power of $2$ .

$y = - 42875 + 18375 x - 105 (25 x^{2}) + {(5 x)}^{3}$

Multiply $25$ by $- 105$ .

$y = - 42875 + 18375 x - 2625 x^{2} + {(5 x)}^{3}$

Apply the product rule to $5 x$ .

$y = - 42875 + 18375 x - 2625 x^{2} + 5^{3} x^{3}$

Raise $5$ to the power of $3$ .

$y = - 42875 + 18375 x - 2625 x^{2} + 125 x^{3}$

Simplify $- 42875 + 18375 x - 2625 x^{2} + 125 x^{3}$ .

Move $- 42875$ .

$y = 18375 x - 2625 x^{2} + 125 x^{3} - 42875$

Move $18375 x$ .

$y = - 2625 x^{2} + 125 x^{3} + 18375 x - 42875$

Reorder $- 2625 x^{2}$ and $125 x^{3}$ .

$y = 125 x^{3} - 2625 x^{2} + 18375 x - 42875$

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

$f^{- 1} (x) = 125 x^{3} - 2625 x^{2} + 18375 x - 42875$

Verify if $f^{- 1} (x) = 125 x^{3} - 2625 x^{2} + 18375 x - 42875$ is the inverse of $f (x) = \frac{x^{\frac{1}{3}}}{5} + 7$ .

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 $f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = 125 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{3} - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Simplify each term.

Use the Binomial Theorem.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = 125 ({(\frac{x^{\frac{1}{3}}}{5})}^{3} + 3 {(\frac{x^{\frac{1}{3}}}{5})}^{2} \cdot 7 + 3 (\frac{x^{\frac{1}{3}}}{5}) \cdot 7^{2} + 7^{3}) - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Simplify each term.

Apply the product rule to $\frac{x^{\frac{1}{3}}}{5}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = 125 (\frac{{(x^{\frac{1}{3}})}^{3}}{5^{3}} + 3 {(\frac{x^{\frac{1}{3}}}{5})}^{2} \cdot 7 + 3 (\frac{x^{\frac{1}{3}}}{5}) \cdot 7^{2} + 7^{3}) - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Simplify the numerator.

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = 125 (\frac{x^{\frac{1}{3} \cdot 3}}{5^{3}} + 3 {(\frac{x^{\frac{1}{3}}}{5})}^{2} \cdot 7 + 3 (\frac{x^{\frac{1}{3}}}{5}) \cdot 7^{2} + 7^{3}) - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Cancel the common factor of $3$ .

Cancel the common factor.

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = 125 (\frac{x}{5^{3}} + 3 {(\frac{x^{\frac{1}{3}}}{5})}^{2} \cdot 7 + 3 (\frac{x^{\frac{1}{3}}}{5}) \cdot 7^{2} + 7^{3}) - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Simplify.

Raise $5$ to the power of $3$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = 125 (\frac{x}{125} + 3 {(\frac{x^{\frac{1}{3}}}{5})}^{2} \cdot 7 + 3 (\frac{x^{\frac{1}{3}}}{5}) \cdot 7^{2} + 7^{3}) - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Apply the product rule to $\frac{x^{\frac{1}{3}}}{5}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = 125 (\frac{x}{125} + 3 (\frac{{(x^{\frac{1}{3}})}^{2}}{5^{2}}) \cdot 7 + 3 (\frac{x^{\frac{1}{3}}}{5}) \cdot 7^{2} + 7^{3}) - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Multiply the exponents in ${(x^{\frac{1}{3}})}^{2}$ .

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

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = 125 (\frac{x}{125} + 3 (\frac{x^{\frac{1}{3} \cdot 2}}{5^{2}}) \cdot 7 + 3 (\frac{x^{\frac{1}{3}}}{5}) \cdot 7^{2} + 7^{3}) - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Combine $\frac{1}{3}$ and $2$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = 125 (\frac{x}{125} + 3 (\frac{x^{\frac{2}{3}}}{5^{2}}) \cdot 7 + 3 (\frac{x^{\frac{1}{3}}}{5}) \cdot 7^{2} + 7^{3}) - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Raise $5$ to the power of $2$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = 125 (\frac{x}{125} + 3 (\frac{x^{\frac{2}{3}}}{25}) \cdot 7 + 3 (\frac{x^{\frac{1}{3}}}{5}) \cdot 7^{2} + 7^{3}) - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Combine $3$ and $\frac{x^{\frac{2}{3}}}{25}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = 125 (\frac{x}{125} + \frac{3 x^{\frac{2}{3}}}{25} \cdot 7 + 3 (\frac{x^{\frac{1}{3}}}{5}) \cdot 7^{2} + 7^{3}) - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Multiply $\frac{3 x^{\frac{2}{3}}}{25} \cdot 7$ .

Combine $\frac{3 x^{\frac{2}{3}}}{25}$ and $7$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = 125 (\frac{x}{125} + \frac{3 x^{\frac{2}{3}} \cdot 7}{25} + 3 (\frac{x^{\frac{1}{3}}}{5}) \cdot 7^{2} + 7^{3}) - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Multiply $7$ by $3$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = 125 (\frac{x}{125} + \frac{21 x^{\frac{2}{3}}}{25} + 3 (\frac{x^{\frac{1}{3}}}{5}) \cdot 7^{2} + 7^{3}) - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Combine $3$ and $\frac{x^{\frac{1}{3}}}{5}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = 125 (\frac{x}{125} + \frac{21 x^{\frac{2}{3}}}{25} + \frac{3 x^{\frac{1}{3}}}{5} \cdot 7^{2} + 7^{3}) - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Raise $7$ to the power of $2$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = 125 (\frac{x}{125} + \frac{21 x^{\frac{2}{3}}}{25} + \frac{3 x^{\frac{1}{3}}}{5} \cdot 49 + 7^{3}) - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Multiply $\frac{3 x^{\frac{1}{3}}}{5} \cdot 49$ .

Combine $\frac{3 x^{\frac{1}{3}}}{5}$ and $49$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = 125 (\frac{x}{125} + \frac{21 x^{\frac{2}{3}}}{25} + \frac{3 x^{\frac{1}{3}} \cdot 49}{5} + 7^{3}) - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Multiply $49$ by $3$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = 125 (\frac{x}{125} + \frac{21 x^{\frac{2}{3}}}{25} + \frac{147 x^{\frac{1}{3}}}{5} + 7^{3}) - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Raise $7$ to the power of $3$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = 125 (\frac{x}{125} + \frac{21 x^{\frac{2}{3}}}{25} + \frac{147 x^{\frac{1}{3}}}{5} + 343) - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Apply the distributive property.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = 125 (\frac{x}{125}) + 125 (\frac{21 x^{\frac{2}{3}}}{25}) + 125 (\frac{147 x^{\frac{1}{3}}}{5}) + 125 \cdot 343 - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Simplify.

Cancel the common factor of $125$ .

Cancel the common factor.

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 125 (\frac{21 x^{\frac{2}{3}}}{25}) + 125 (\frac{147 x^{\frac{1}{3}}}{5}) + 125 \cdot 343 - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Cancel the common factor of $25$ .

Factor $25$ out of $125$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 25 (5) (\frac{21 x^{\frac{2}{3}}}{25}) + 125 (\frac{147 x^{\frac{1}{3}}}{5}) + 125 \cdot 343 - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Cancel the common factor.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 25 \cdot (5 (\frac{21 x^{\frac{2}{3}}}{25})) + 125 (\frac{147 x^{\frac{1}{3}}}{5}) + 125 \cdot 343 - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 5 (21 x^{\frac{2}{3}}) + 125 (\frac{147 x^{\frac{1}{3}}}{5}) + 125 \cdot 343 - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Multiply $21$ by $5$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 125 (\frac{147 x^{\frac{1}{3}}}{5}) + 125 \cdot 343 - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Cancel the common factor of $5$ .

Factor $5$ out of $125$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 5 (25) (\frac{147 x^{\frac{1}{3}}}{5}) + 125 \cdot 343 - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Cancel the common factor.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 5 \cdot (25 (\frac{147 x^{\frac{1}{3}}}{5})) + 125 \cdot 343 - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 25 (147 x^{\frac{1}{3}}) + 125 \cdot 343 - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Multiply $147$ by $25$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 125 \cdot 343 - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Multiply $125$ by $343$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 2625 {(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2} + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Rewrite ${(\frac{x^{\frac{1}{3}}}{5} + 7)}^{2}$ as $(\frac{x^{\frac{1}{3}}}{5} + 7) (\frac{x^{\frac{1}{3}}}{5} + 7)$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 2625 ((\frac{x^{\frac{1}{3}}}{5} + 7) (\frac{x^{\frac{1}{3}}}{5} + 7)) + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Expand $(\frac{x^{\frac{1}{3}}}{5} + 7) (\frac{x^{\frac{1}{3}}}{5} + 7)$ using the FOIL Method.

Apply the distributive property.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 2625 (\frac{x^{\frac{1}{3}}}{5} \cdot (\frac{x^{\frac{1}{3}}}{5} + 7) + 7 (\frac{x^{\frac{1}{3}}}{5} + 7)) + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Apply the distributive property.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 2625 (\frac{x^{\frac{1}{3}}}{5} \cdot \frac{x^{\frac{1}{3}}}{5} + \frac{x^{\frac{1}{3}}}{5} \cdot 7 + 7 (\frac{x^{\frac{1}{3}}}{5} + 7)) + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Apply the distributive property.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 2625 (\frac{x^{\frac{1}{3}}}{5} \cdot \frac{x^{\frac{1}{3}}}{5} + \frac{x^{\frac{1}{3}}}{5} \cdot 7 + 7 (\frac{x^{\frac{1}{3}}}{5}) + 7 \cdot 7) + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Simplify and combine like terms.

Simplify each term.

Combine.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 2625 (\frac{x^{\frac{1}{3}} x^{\frac{1}{3}}}{5 \cdot 5} + \frac{x^{\frac{1}{3}}}{5} \cdot 7 + 7 (\frac{x^{\frac{1}{3}}}{5}) + 7 \cdot 7) + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{1}{3}}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 2625 (\frac{x^{\frac{1}{3} + \frac{1}{3}}}{5 \cdot 5} + \frac{x^{\frac{1}{3}}}{5} \cdot 7 + 7 (\frac{x^{\frac{1}{3}}}{5}) + 7 \cdot 7) + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 2625 (\frac{x^{\frac{1 + 1}{3}}}{5 \cdot 5} + \frac{x^{\frac{1}{3}}}{5} \cdot 7 + 7 (\frac{x^{\frac{1}{3}}}{5}) + 7 \cdot 7) + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Add $1$ and $1$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 2625 (\frac{x^{\frac{2}{3}}}{5 \cdot 5} + \frac{x^{\frac{1}{3}}}{5} \cdot 7 + 7 (\frac{x^{\frac{1}{3}}}{5}) + 7 \cdot 7) + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Multiply $5$ by $5$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 2625 (\frac{x^{\frac{2}{3}}}{25} + \frac{x^{\frac{1}{3}}}{5} \cdot 7 + 7 (\frac{x^{\frac{1}{3}}}{5}) + 7 \cdot 7) + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Combine $\frac{x^{\frac{1}{3}}}{5}$ and $7$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 2625 (\frac{x^{\frac{2}{3}}}{25} + \frac{x^{\frac{1}{3}} \cdot 7}{5} + 7 (\frac{x^{\frac{1}{3}}}{5}) + 7 \cdot 7) + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Move $7$ to the left of $x^{\frac{1}{3}}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 2625 (\frac{x^{\frac{2}{3}}}{25} + \frac{7 \cdot x^{\frac{1}{3}}}{5} + 7 (\frac{x^{\frac{1}{3}}}{5}) + 7 \cdot 7) + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Combine $7$ and $\frac{x^{\frac{1}{3}}}{5}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 2625 (\frac{x^{\frac{2}{3}}}{25} + \frac{7 x^{\frac{1}{3}}}{5} + \frac{7 x^{\frac{1}{3}}}{5} + 7 \cdot 7) + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Multiply $7$ by $7$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 2625 (\frac{x^{\frac{2}{3}}}{25} + \frac{7 x^{\frac{1}{3}}}{5} + \frac{7 x^{\frac{1}{3}}}{5} + 49) + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Add $\frac{7 x^{\frac{1}{3}}}{5}$ and $\frac{7 x^{\frac{1}{3}}}{5}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 2625 (\frac{x^{\frac{2}{3}}}{25} + 2 (\frac{7 x^{\frac{1}{3}}}{5}) + 49) + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Multiply $2 \frac{7 x^{\frac{1}{3}}}{5}$ .

Combine $2$ and $\frac{7 x^{\frac{1}{3}}}{5}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 2625 (\frac{x^{\frac{2}{3}}}{25} + \frac{2 (7 x^{\frac{1}{3}})}{5} + 49) + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Multiply $7$ by $2$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 2625 (\frac{x^{\frac{2}{3}}}{25} + \frac{14 x^{\frac{1}{3}}}{5} + 49) + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Apply the distributive property.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 2625 \frac{x^{\frac{2}{3}}}{25} - 2625 \frac{14 x^{\frac{1}{3}}}{5} - 2625 \cdot 49 + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Simplify.

Cancel the common factor of $25$ .

Factor $25$ out of $- 2625$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 + 25 (- 105) (\frac{x^{\frac{2}{3}}}{25}) - 2625 \frac{14 x^{\frac{1}{3}}}{5} - 2625 \cdot 49 + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Cancel the common factor.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 + 25 \cdot (- 105 \frac{x^{\frac{2}{3}}}{25}) - 2625 \frac{14 x^{\frac{1}{3}}}{5} - 2625 \cdot 49 + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 105 x^{\frac{2}{3}} - 2625 \frac{14 x^{\frac{1}{3}}}{5} - 2625 \cdot 49 + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Cancel the common factor of $5$ .

Factor $5$ out of $- 2625$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 105 x^{\frac{2}{3}} + 5 (- 525) (\frac{14 x^{\frac{1}{3}}}{5}) - 2625 \cdot 49 + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Cancel the common factor.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 105 x^{\frac{2}{3}} + 5 \cdot (- 525 \frac{14 x^{\frac{1}{3}}}{5}) - 2625 \cdot 49 + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 105 x^{\frac{2}{3}} - 525 (14 x^{\frac{1}{3}}) - 2625 \cdot 49 + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Multiply $14$ by $- 525$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 105 x^{\frac{2}{3}} - 7350 x^{\frac{1}{3}} - 2625 \cdot 49 + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Multiply $- 2625$ by $49$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 105 x^{\frac{2}{3}} - 7350 x^{\frac{1}{3}} - 128625 + 18375 (\frac{x^{\frac{1}{3}}}{5} + 7) - 42875$

Apply the distributive property.

Cancel the common factor of $5$ .

Factor $5$ out of $18375$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 105 x^{\frac{2}{3}} - 7350 x^{\frac{1}{3}} - 128625 + 5 (3675) (\frac{x^{\frac{1}{3}}}{5}) + 18375 \cdot 7 - 42875$

Cancel the common factor.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 105 x^{\frac{2}{3}} - 7350 x^{\frac{1}{3}} - 128625 + 5 \cdot (3675 (\frac{x^{\frac{1}{3}}}{5})) + 18375 \cdot 7 - 42875$

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 105 x^{\frac{2}{3}} - 7350 x^{\frac{1}{3}} - 128625 + 3675 x^{\frac{1}{3}} + 18375 \cdot 7 - 42875$

Multiply $18375$ by $7$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 105 x^{\frac{2}{3}} - 7350 x^{\frac{1}{3}} - 128625 + 3675 x^{\frac{1}{3}} + 128625 - 42875$

Simplify by adding terms.

Combine the opposite terms in $x + 105 x^{\frac{2}{3}} + 3675 x^{\frac{1}{3}} + 42875 - 105 x^{\frac{2}{3}} - 7350 x^{\frac{1}{3}} - 128625 + 3675 x^{\frac{1}{3}} + 128625 - 42875$ .

Subtract $105 x^{\frac{2}{3}}$ from $105 x^{\frac{2}{3}}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 3675 x^{\frac{1}{3}} + 42875 + 0 - 7350 x^{\frac{1}{3}} - 128625 + 3675 x^{\frac{1}{3}} + 128625 - 42875$

Add $x + 3675 x^{\frac{1}{3}} + 42875$ and $0$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 3675 x^{\frac{1}{3}} + 42875 - 7350 x^{\frac{1}{3}} - 128625 + 3675 x^{\frac{1}{3}} + 128625 - 42875$

Add $- 128625$ and $128625$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 3675 x^{\frac{1}{3}} + 42875 - 7350 x^{\frac{1}{3}} + 3675 x^{\frac{1}{3}} + 0 - 42875$

Add $x + 3675 x^{\frac{1}{3}} + 42875 - 7350 x^{\frac{1}{3}} + 3675 x^{\frac{1}{3}}$ and $0$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 3675 x^{\frac{1}{3}} + 42875 - 7350 x^{\frac{1}{3}} + 3675 x^{\frac{1}{3}} - 42875$

Subtract $42875$ from $42875$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 3675 x^{\frac{1}{3}} - 7350 x^{\frac{1}{3}} + 3675 x^{\frac{1}{3}} + 0$

Add $x + 3675 x^{\frac{1}{3}} - 7350 x^{\frac{1}{3}} + 3675 x^{\frac{1}{3}}$ and $0$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 3675 x^{\frac{1}{3}} - 7350 x^{\frac{1}{3}} + 3675 x^{\frac{1}{3}}$

Subtract $7350 x^{\frac{1}{3}}$ from $3675 x^{\frac{1}{3}}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x - 3675 x^{\frac{1}{3}} + 3675 x^{\frac{1}{3}}$

Combine the opposite terms in $x - 3675 x^{\frac{1}{3}} + 3675 x^{\frac{1}{3}}$ .

Add $- 3675 x^{\frac{1}{3}}$ and $3675 x^{\frac{1}{3}}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 7) = x$

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

Set up the composite result function.

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

Evaluate $f (125 x^{3} - 2625 x^{2} + 18375 x - 42875)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (125 x^{3} - 2625 x^{2} + 18375 x - 42875) = \frac{{(125 x^{3} - 2625 x^{2} + 18375 x - 42875)}^{\frac{1}{3}}}{5} + 7$

To write $7$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (125 x^{3} - 2625 x^{2} + 18375 x - 42875) = \frac{{(125 x^{3} - 2625 x^{2} + 18375 x - 42875)}^{\frac{1}{3}}}{5} + 7 \cdot \frac{5}{5}$

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

$f (125 x^{3} - 2625 x^{2} + 18375 x - 42875) = \frac{{(125 x^{3} - 2625 x^{2} + 18375 x - 42875)}^{\frac{1}{3}}}{5} + \frac{7 \cdot 5}{5}$

Combine the numerators over the common denominator.

$f (125 x^{3} - 2625 x^{2} + 18375 x - 42875) = \frac{{(125 x^{3} - 2625 x^{2} + 18375 x - 42875)}^{\frac{1}{3}} + 7 \cdot 5}{5}$

Multiply $7$ by $5$ .

$f (125 x^{3} - 2625 x^{2} + 18375 x - 42875) = \frac{{(125 x^{3} - 2625 x^{2} + 18375 x - 42875)}^{\frac{1}{3}} + 35}{5}$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 125 x^{3} - 2625 x^{2} + 18375 x - 42875$ is the inverse of $f (x) = \frac{x^{\frac{1}{3}}}{5} + 7$ .

$f^{- 1} (x) = 125 x^{3} - 2625 x^{2} + 18375 x - 42875$

Find the Inverse f(x)=(x^(1/3))/5+7

Related Questions

Support LCMGCF.com by disabling your adblocker.