Find the Inverse f(x)=4x^5-2

$f (x) = 4 x^{5} - 2$

Step 1

Write $f (x) = 4 x^{5} - 2$ as an equation.

$y = 4 x^{5} - 2$

Step 2

Interchange the variables.

$x = 4 y^{5} - 2$

Step 3

Solve for $y$ .

Rewrite the equation as $4 y^{5} - 2 = x$ .

$4 y^{5} - 2 = x$

Add $2$ to both sides of the equation.

$4 y^{5} = x + 2$

Divide each term in $4 y^{5} = x + 2$ by $4$ and simplify.

Divide each term in $4 y^{5} = x + 2$ by $4$ .

$\frac{4 y^{5}}{4} = \frac{x}{4} + \frac{2}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 y^{5}}{4} = \frac{x}{4} + \frac{2}{4}$

Divide $y^{5}$ by $1$ .

$y^{5} = \frac{x}{4} + \frac{2}{4}$

Simplify the right side.

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2$ .

$y^{5} = \frac{x}{4} + \frac{2 (1)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$y^{5} = \frac{x}{4} + \frac{2 \cdot 1}{2 \cdot 2}$

Cancel the common factor.

$y^{5} = \frac{x}{4} + \frac{2 \cdot 1}{2 \cdot 2}$

Rewrite the expression.

$y^{5} = \frac{x}{4} + \frac{1}{2}$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[5]{\frac{x}{4} + \frac{1}{2}}$

Simplify $\sqrt[5]{\frac{x}{4} + \frac{1}{2}}$ .

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \sqrt[5]{\frac{x}{4} + \frac{1}{2} \cdot \frac{2}{2}}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

$y = \sqrt[5]{\frac{x}{4} + \frac{2}{2 \cdot 2}}$

Multiply $2$ by $2$ .

$y = \sqrt[5]{\frac{x}{4} + \frac{2}{4}}$

Combine the numerators over the common denominator.

$y = \sqrt[5]{\frac{x + 2}{4}}$

Rewrite $\sqrt[5]{\frac{x + 2}{4}}$ as $\frac{\sqrt[5]{x + 2}}{\sqrt[5]{4}}$ .

$y = \frac{\sqrt[5]{x + 2}}{\sqrt[5]{4}}$

Multiply $\frac{\sqrt[5]{x + 2}}{\sqrt[5]{4}}$ by $\frac{{\sqrt[5]{4}}^{4}}{{\sqrt[5]{4}}^{4}}$ .

$y = \frac{\sqrt[5]{x + 2}}{\sqrt[5]{4}} \cdot \frac{{\sqrt[5]{4}}^{4}}{{\sqrt[5]{4}}^{4}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt[5]{x + 2}}{\sqrt[5]{4}}$ by $\frac{{\sqrt[5]{4}}^{4}}{{\sqrt[5]{4}}^{4}}$ .

$y = \frac{\sqrt[5]{x + 2} {\sqrt[5]{4}}^{4}}{\sqrt[5]{4} {\sqrt[5]{4}}^{4}}$

Raise $\sqrt[5]{4}$ to the power of $1$ .

$y = \frac{\sqrt[5]{x + 2} {\sqrt[5]{4}}^{4}}{{\sqrt[5]{4}}^{1} {\sqrt[5]{4}}^{4}}$

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

$y = \frac{\sqrt[5]{x + 2} {\sqrt[5]{4}}^{4}}{{\sqrt[5]{4}}^{1 + 4}}$

Add $1$ and $4$ .

$y = \frac{\sqrt[5]{x + 2} {\sqrt[5]{4}}^{4}}{{\sqrt[5]{4}}^{5}}$

Rewrite ${\sqrt[5]{4}}^{5}$ as $4$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{4}$ as $4^{\frac{1}{5}}$ .

$y = \frac{\sqrt[5]{x + 2} {\sqrt[5]{4}}^{4}}{{(4^{\frac{1}{5}})}^{5}}$

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

$y = \frac{\sqrt[5]{x + 2} {\sqrt[5]{4}}^{4}}{4^{\frac{1}{5} \cdot 5}}$

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

$y = \frac{\sqrt[5]{x + 2} {\sqrt[5]{4}}^{4}}{4^{\frac{5}{5}}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$y = \frac{\sqrt[5]{x + 2} {\sqrt[5]{4}}^{4}}{4^{\frac{5}{5}}}$

Rewrite the expression.

$y = \frac{\sqrt[5]{x + 2} {\sqrt[5]{4}}^{4}}{4^{1}}$

Evaluate the exponent.

$y = \frac{\sqrt[5]{x + 2} {\sqrt[5]{4}}^{4}}{4}$

Simplify the numerator.

Rewrite ${\sqrt[5]{4}}^{4}$ as $\sqrt[5]{4^{4}}$ .

$y = \frac{\sqrt[5]{x + 2} \sqrt[5]{4^{4}}}{4}$

Raise $4$ to the power of $4$ .

$y = \frac{\sqrt[5]{x + 2} \sqrt[5]{256}}{4}$

Rewrite $256$ as $2^{5} \cdot 8$ .

Factor $32$ out of $256$ .

$y = \frac{\sqrt[5]{x + 2} \sqrt[5]{32 (8)}}{4}$

Rewrite $32$ as $2^{5}$ .

$y = \frac{\sqrt[5]{x + 2} \sqrt[5]{2^{5} \cdot 8}}{4}$

Pull terms out from under the radical.

$y = \frac{\sqrt[5]{x + 2} \cdot 2 \sqrt[5]{8}}{4}$

Combine using the product rule for radicals.

$y = \frac{2 \sqrt[5]{(x + 2) \cdot 8}}{4}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2 \sqrt[5]{(x + 2) \cdot 8}$ .

$y = \frac{2 (\sqrt[5]{(x + 2) \cdot 8})}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$y = \frac{2 \sqrt[5]{(x + 2) \cdot 8}}{2 \cdot 2}$

Cancel the common factor.

$y = \frac{2 \sqrt[5]{(x + 2) \cdot 8}}{2 \cdot 2}$

Rewrite the expression.

$y = \frac{\sqrt[5]{(x + 2) \cdot 8}}{2}$

Reorder factors in $\frac{\sqrt[5]{(x + 2) \cdot 8}}{2}$ .

$y = \frac{\sqrt[5]{8 (x + 2)}}{2}$

Step 4

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

$f^{- 1} (x) = \frac{\sqrt[5]{8 (x + 2)}}{2}$

Step 5

Verify if $f^{- 1} (x) = \frac{\sqrt[5]{8 (x + 2)}}{2}$ is the inverse of $f (x) = 4 x^{5} - 2$ .

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 $f^{- 1} (4 x^{5} - 2)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (4 x^{5} - 2) = \frac{\sqrt[5]{8 ((4 x^{5} - 2) + 2)}}{2}$

Simplify the numerator.

Add $- 2$ and $2$ .

$f^{- 1} (4 x^{5} - 2) = \frac{\sqrt[5]{8 (4 x^{5} + 0)}}{2}$

Add $4 x^{5}$ and $0$ .

$f^{- 1} (4 x^{5} - 2) = \frac{\sqrt[5]{8 \cdot (4 x^{5})}}{2}$

Multiply $8$ by $4$ .

$f^{- 1} (4 x^{5} - 2) = \frac{\sqrt[5]{32 x^{5}}}{2}$

Rewrite $32 x^{5}$ as ${(2 x)}^{5}$ .

$f^{- 1} (4 x^{5} - 2) = \frac{\sqrt[5]{{(2 x)}^{5}}}{2}$

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

$f^{- 1} (4 x^{5} - 2) = \frac{2 x}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (4 x^{5} - 2) = \frac{2 x}{2}$

Divide $x$ by $1$ .

$f^{- 1} (4 x^{5} - 2) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{\sqrt[5]{8 (x + 2)}}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\sqrt[5]{8 (x + 2)}}{2}) = 4 {(\frac{\sqrt[5]{8 (x + 2)}}{2})}^{5} - 2$

Simplify each term.

Apply the product rule to $\frac{\sqrt[5]{8 (x + 2)}}{2}$ .

$f (\frac{\sqrt[5]{8 (x + 2)}}{2}) = 4 (\frac{{\sqrt[5]{8 (x + 2)}}^{5}}{2^{5}}) - 2$

Simplify the numerator.

Rewrite ${\sqrt[5]{8 (x + 2)}}^{5}$ as $8 (x + 2)$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{8 (x + 2)}$ as ${(8 (x + 2))}^{\frac{1}{5}}$ .

$f (\frac{\sqrt[5]{8 (x + 2)}}{2}) = 4 (\frac{{({(8 (x + 2))}^{\frac{1}{5}})}^{5}}{2^{5}}) - 2$

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

$f (\frac{\sqrt[5]{8 (x + 2)}}{2}) = 4 (\frac{{(8 (x + 2))}^{\frac{1}{5} \cdot 5}}{2^{5}}) - 2$

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

$f (\frac{\sqrt[5]{8 (x + 2)}}{2}) = 4 (\frac{{(8 (x + 2))}^{\frac{5}{5}}}{2^{5}}) - 2$

Cancel the common factor of $5$ .

Cancel the common factor.

$f (\frac{\sqrt[5]{8 (x + 2)}}{2}) = 4 (\frac{{(8 (x + 2))}^{\frac{5}{5}}}{2^{5}}) - 2$

Rewrite the expression.

$f (\frac{\sqrt[5]{8 (x + 2)}}{2}) = 4 (\frac{8 (x + 2)}{2^{5}}) - 2$

Simplify.

$f (\frac{\sqrt[5]{8 (x + 2)}}{2}) = 4 (\frac{8 (x + 2)}{2^{5}}) - 2$

Apply the distributive property.

$f (\frac{\sqrt[5]{8 (x + 2)}}{2}) = 4 (\frac{8 x + 8 \cdot 2}{2^{5}}) - 2$

Multiply $8$ by $2$ .

$f (\frac{\sqrt[5]{8 (x + 2)}}{2}) = 4 (\frac{8 x + 16}{2^{5}}) - 2$

Factor $8$ out of $8 x + 16$ .

Factor $8$ out of $8 x$ .

$f (\frac{\sqrt[5]{8 (x + 2)}}{2}) = 4 (\frac{8 (x) + 16}{2^{5}}) - 2$

Factor $8$ out of $16$ .

$f (\frac{\sqrt[5]{8 (x + 2)}}{2}) = 4 (\frac{8 x + 8 \cdot 2}{2^{5}}) - 2$

Factor $8$ out of $8 x + 8 \cdot 2$ .

$f (\frac{\sqrt[5]{8 (x + 2)}}{2}) = 4 (\frac{8 (x + 2)}{2^{5}}) - 2$

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

$f (\frac{\sqrt[5]{8 (x + 2)}}{2}) = 4 (\frac{8 (x + 2)}{32}) - 2$

Cancel the common factor of $4$ .

Factor $4$ out of $32$ .

$f (\frac{\sqrt[5]{8 (x + 2)}}{2}) = 4 (\frac{8 (x + 2)}{4 (8)}) - 2$

Cancel the common factor.

$f (\frac{\sqrt[5]{8 (x + 2)}}{2}) = 4 (\frac{8 (x + 2)}{4 \cdot 8}) - 2$

Rewrite the expression.

$f (\frac{\sqrt[5]{8 (x + 2)}}{2}) = \frac{8 (x + 2)}{8} - 2$

Cancel the common factor of $8$ .

Cancel the common factor.

$f (\frac{\sqrt[5]{8 (x + 2)}}{2}) = \frac{8 (x + 2)}{8} - 2$

Divide $x + 2$ by $1$ .

$f (\frac{\sqrt[5]{8 (x + 2)}}{2}) = x + 2 - 2$

Combine the opposite terms in $x + 2 - 2$ .

Subtract $2$ from $2$ .

$f (\frac{\sqrt[5]{8 (x + 2)}}{2}) = x + 0$

Add $x$ and $0$ .

$f (\frac{\sqrt[5]{8 (x + 2)}}{2}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\sqrt[5]{8 (x + 2)}}{2}$ is the inverse of $f (x) = 4 x^{5} - 2$ .

$f^{- 1} (x) = \frac{\sqrt[5]{8 (x + 2)}}{2}$

Find the Inverse f(x)=4x^5-2

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