Trigonometry Examples

$f (x) = 7 x^{9}$

Write $f (x) = 7 x^{9}$ as an equation.

$y = 7 x^{9}$

Interchange the variables.

$x = 7 y^{9}$

Solve for $y$ .

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Rewrite the equation as $7 y^{9} = x$ .

$7 y^{9} = x$

Divide each term in $7 y^{9} = x$ by $7$ and simplify.

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Divide each term in $7 y^{9} = x$ by $7$ .

$\frac{7 y^{9}}{7} = \frac{x}{7}$

Simplify the left side.

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Cancel the common factor of $7$ .

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Cancel the common factor.

$\frac{7 y^{9}}{7} = \frac{x}{7}$

Divide $y^{9}$ by $1$ .

$y^{9} = \frac{x}{7}$

Take the 9th root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[9]{\frac{x}{7}}$

Rewrite $\sqrt[9]{\frac{x}{7}}$ as $\frac{\sqrt[9]{x}}{\sqrt[9]{7}}$ .

$y = \frac{\sqrt[9]{x}}{\sqrt[9]{7}}$

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

$f^{- 1} (x) = \frac{\sqrt[9]{x}}{\sqrt[9]{7}}$

Verify if $f^{- 1} (x) = \frac{\sqrt[9]{x}}{\sqrt[9]{7}}$ is the inverse of $f (x) = 7 x^{9}$ .

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

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

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Set up the composite result function.

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

Evaluate $f^{- 1} (7 x^{9})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (7 x^{9}) = \frac{\sqrt[9]{7 x^{9}}}{\sqrt[9]{7}}$

Combine $\sqrt[9]{7 x^{9}}$ and $\sqrt[9]{7}$ into a single radical.

$f^{- 1} (7 x^{9}) = \sqrt[9]{\frac{7 x^{9}}{7}}$

Reduce the expression by cancelling the common factors.

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Reduce the expression $\frac{7 x^{9}}{7}$ by cancelling the common factors.

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Cancel the common factor.

$f^{- 1} (7 x^{9}) = \sqrt[9]{\frac{7 x^{9}}{7}}$

Rewrite the expression.

$f^{- 1} (7 x^{9}) = \sqrt[9]{\frac{x^{9}}{1}}$

Divide $x^{9}$ by $1$ .

$f^{- 1} (7 x^{9}) = \sqrt[9]{x^{9}}$

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

$f^{- 1} (7 x^{9}) = x$

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

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Set up the composite result function.

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

Evaluate $f (\frac{\sqrt[9]{x}}{\sqrt[9]{7}})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\sqrt[9]{x}}{\sqrt[9]{7}}) = 7 {(\frac{\sqrt[9]{x}}{\sqrt[9]{7}})}^{9}$

Apply the product rule to $\frac{\sqrt[9]{x}}{\sqrt[9]{7}}$ .

$f (\frac{\sqrt[9]{x}}{\sqrt[9]{7}}) = 7 (\frac{{\sqrt[9]{x}}^{9}}{{\sqrt[9]{7}}^{9}})$

Rewrite ${\sqrt[9]{x}}^{9}$ as $x$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[9]{x}$ as $x^{\frac{1}{9}}$ .

$f (\frac{\sqrt[9]{x}}{\sqrt[9]{7}}) = 7 (\frac{{(x^{\frac{1}{9}})}^{9}}{{\sqrt[9]{7}}^{9}})$

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

$f (\frac{\sqrt[9]{x}}{\sqrt[9]{7}}) = 7 (\frac{x^{\frac{1}{9} \cdot 9}}{{\sqrt[9]{7}}^{9}})$

Combine $\frac{1}{9}$ and $9$ .

$f (\frac{\sqrt[9]{x}}{\sqrt[9]{7}}) = 7 (\frac{x^{\frac{9}{9}}}{{\sqrt[9]{7}}^{9}})$

Cancel the common factor of $9$ .

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Cancel the common factor.

$f (\frac{\sqrt[9]{x}}{\sqrt[9]{7}}) = 7 (\frac{x^{\frac{9}{9}}}{{\sqrt[9]{7}}^{9}})$

Rewrite the expression.

$f (\frac{\sqrt[9]{x}}{\sqrt[9]{7}}) = 7 (\frac{x}{{\sqrt[9]{7}}^{9}})$

Simplify.

$f (\frac{\sqrt[9]{x}}{\sqrt[9]{7}}) = 7 (\frac{x}{{\sqrt[9]{7}}^{9}})$

Rewrite ${\sqrt[9]{7}}^{9}$ as $7$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[9]{7}$ as $7^{\frac{1}{9}}$ .

$f (\frac{\sqrt[9]{x}}{\sqrt[9]{7}}) = 7 (\frac{x}{{(7^{\frac{1}{9}})}^{9}})$

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

$f (\frac{\sqrt[9]{x}}{\sqrt[9]{7}}) = 7 (\frac{x}{7^{\frac{1}{9} \cdot 9}})$

Combine $\frac{1}{9}$ and $9$ .

$f (\frac{\sqrt[9]{x}}{\sqrt[9]{7}}) = 7 (\frac{x}{7^{\frac{9}{9}}})$

Cancel the common factor of $9$ .

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Cancel the common factor.

$f (\frac{\sqrt[9]{x}}{\sqrt[9]{7}}) = 7 (\frac{x}{7^{\frac{9}{9}}})$

Rewrite the expression.

$f (\frac{\sqrt[9]{x}}{\sqrt[9]{7}}) = 7 (\frac{x}{7})$

Evaluate the exponent.

$f (\frac{\sqrt[9]{x}}{\sqrt[9]{7}}) = 7 (\frac{x}{7})$

Cancel the common factor of $7$ .

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Cancel the common factor.

$f (\frac{\sqrt[9]{x}}{\sqrt[9]{7}}) = 7 (\frac{x}{7})$

Rewrite the expression.

$f (\frac{\sqrt[9]{x}}{\sqrt[9]{7}}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\sqrt[9]{x}}{\sqrt[9]{7}}$ is the inverse of $f (x) = 7 x^{9}$ .

$f^{- 1} (x) = \frac{\sqrt[9]{x}}{\sqrt[9]{7}}$