Find the Inverse f(x)=(x^4)/7+27

$f (x) = \frac{x^{4}}{7} + 27$

Replace $f (x)$ with $y$ .

$y = \frac{x^{4}}{7} + 27$

Interchange the variables.

$x = \frac{y^{4}}{7} + 27$

Solve for $y$ .

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Rewrite the equation as $\frac{y^{4}}{7} + 27 = x$ .

$\frac{y^{4}}{7} + 27 = x$

Subtract $27$ from both sides of the equation.

$\frac{y^{4}}{7} = x - 27$

Multiply both sides of the equation by $7$ .

$7 \cdot \frac{y^{4}}{7} = 7 \cdot (x - 27)$

Simplify both sides of the equation.

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Cancel the common factor of $7$ .

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Cancel the common factor.

$7 \cdot \frac{y^{4}}{7} = 7 \cdot (x - 27)$

Rewrite the expression.

$y^{4} = 7 \cdot (x - 27)$

Simplify $7 \cdot (x - 27)$ .

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Apply the distributive property.

$y^{4} = 7 x + 7 \cdot - 27$

Multiply $7$ by $- 27$ .

$y^{4} = 7 x - 189$

Take the 4th root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt[4]{7 x - 189}$

The complete solution is the result of both the positive and negative portions of the solution.

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Factor $7$ out of $7 x - 189$ .

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Factor $7$ out of $7 x$ .

$y = \pm \sqrt[4]{7 (x) - 189}$

Factor $7$ out of $- 189$ .

$y = \pm \sqrt[4]{7 x + 7 \cdot - 27}$

Factor $7$ out of $7 x + 7 \cdot - 27$ .

$y = \pm \sqrt[4]{7 (x - 27)}$

The complete solution is the result of both the positive and negative portions of the solution.

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First, use the positive value of the $\pm$ to find the first solution.

$y = \sqrt[4]{7 (x - 27)}$

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \sqrt[4]{7 (x - 27)}$

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt[4]{7 (x - 27)}$

$y = - \sqrt[4]{7 (x - 27)}$

$y = \sqrt[4]{7 (x - 27)}$

$y = - \sqrt[4]{7 (x - 27)}$

$y = \sqrt[4]{7 (x - 27)}$

$y = - \sqrt[4]{7 (x - 27)}$

$y = \sqrt[4]{7 (x - 27)}$

$y = - \sqrt[4]{7 (x - 27)}$

Solve for $y$ and replace with $f^{- 1} (x)$ .

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

$f^{- 1} (x) = \sqrt[4]{7 (x - 27)}, - \sqrt[4]{7 (x - 27)}$

Set up the composite result function.

$g (f (x))$

Evaluate $g (f (x))$ by substituting in the value of $f$ into $g$ .

$g (\frac{x^{4}}{7} + 27) = \sqrt[4]{7 ((\frac{x^{4}}{7} + 27) - 27)}$

Simplify by subtracting numbers.

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Subtract $27$ from $27$ .

$g (\frac{x^{4}}{7} + 27) = \sqrt[4]{7 (\frac{x^{4}}{7} + 0)}$

Add $\frac{x^{4}}{7}$ and $0$ .

$g (\frac{x^{4}}{7} + 27) = \sqrt[4]{7 (\frac{x^{4}}{7})}$

Combine $7$ and $\frac{x^{4}}{7}$ .

$g (\frac{x^{4}}{7} + 27) = \sqrt[4]{\frac{7 x^{4}}{7}}$

Reduce the expression by cancelling the common factors.

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Reduce the expression $\frac{7 x^{4}}{7}$ by cancelling the common factors.

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Cancel the common factor.

$g (\frac{x^{4}}{7} + 27) = \sqrt[4]{\frac{7 x^{4}}{7}}$

Rewrite the expression.

$g (\frac{x^{4}}{7} + 27) = \sqrt[4]{\frac{x^{4}}{1}}$

Divide $x^{4}$ by $1$ .

$g (\frac{x^{4}}{7} + 27) = \sqrt[4]{x^{4}}$

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

$g (\frac{x^{4}}{7} + 27) = x$

Since $g (f (x)) = x$ , $f^{- 1} (x) = \sqrt[4]{7 (x - 27)}, - \sqrt[4]{7 (x - 27)}$ is the inverse of $f (x) = \frac{x^{4}}{7} + 27$ .

$f^{- 1} (x) = \sqrt[4]{7 (x - 27)}, - \sqrt[4]{7 (x - 27)}$

Find the Inverse f(x)=(x^4)/7+27

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