Trigonometry Examples

$- 9 \sqrt{x}$

Step 1

Interchange the variables.

$x = - 9 \sqrt{y}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $- 9 \sqrt{y} = x$ .

$- 9 \sqrt{y} = x$

Step 2.2

Divide each term in $- 9 \sqrt{y} = x$ by $- 9$ and simplify.

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Step 2.2.1

Divide each term in $- 9 \sqrt{y} = x$ by $- 9$ .

$\frac{- 9 \sqrt{y}}{- 9} = \frac{x}{- 9}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 9$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 9 \sqrt{y}}{- 9} = \frac{x}{- 9}$

Step 2.2.2.1.2

Divide $\sqrt{y}$ by $1$ .

$\sqrt{y} = \frac{x}{- 9}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Move the negative in front of the fraction.

$\sqrt{y} = - \frac{x}{9}$

Step 2.3

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{y}}^{2} = {(- \frac{x}{9})}^{2}$

Step 2.4

Simplify each side of the equation.

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Step 2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

${(y^{\frac{1}{2}})}^{2} = {(- \frac{x}{9})}^{2}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Simplify ${(y^{\frac{1}{2}})}^{2}$ .

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Step 2.4.2.1.1

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

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Step 2.4.2.1.1.1

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

$y^{\frac{1}{2} \cdot 2} = {(- \frac{x}{9})}^{2}$

Step 2.4.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.4.2.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{2} \cdot 2} = {(- \frac{x}{9})}^{2}$

Step 2.4.2.1.1.2.2

Rewrite the expression.

$y^{1} = {(- \frac{x}{9})}^{2}$

Step 2.4.2.1.2

Simplify.

$y = {(- \frac{x}{9})}^{2}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Simplify ${(- \frac{x}{9})}^{2}$ .

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Step 2.4.3.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.4.3.1.1.1

Apply the product rule to $- \frac{x}{9}$ .

$y = {(- 1)}^{2} {(\frac{x}{9})}^{2}$

Step 2.4.3.1.1.2

Apply the product rule to $\frac{x}{9}$ .

$y = {(- 1)}^{2} \frac{x^{2}}{9^{2}}$

Step 2.4.3.1.2

Raise $- 1$ to the power of $2$ .

$y = 1 \frac{x^{2}}{9^{2}}$

Step 2.4.3.1.3

Multiply $\frac{x^{2}}{9^{2}}$ by $1$ .

$y = \frac{x^{2}}{9^{2}}$

Step 2.4.3.1.4

Raise $9$ to the power of $2$ .

$y = \frac{x^{2}}{81}$

Step 3

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

$f^{- 1} (x) = \frac{x^{2}}{81}$

Step 4

Verify if $f^{- 1} (x) = \frac{x^{2}}{81}$ is the inverse of $f (x) = - 9 \sqrt{x}$ .

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

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

$f^{- 1} (- 9 \sqrt{x}) = \frac{{(- 9 \sqrt{x})}^{2}}{81}$

Step 4.2.3

Simplify the numerator.

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Step 4.2.3.1

Apply the product rule to $- 9 \sqrt{x}$ .

$f^{- 1} (- 9 \sqrt{x}) = \frac{{(- 9)}^{2} {\sqrt{x}}^{2}}{81}$

Step 4.2.3.2

Raise $- 9$ to the power of $2$ .

$f^{- 1} (- 9 \sqrt{x}) = \frac{81 {\sqrt{x}}^{2}}{81}$

Step 4.2.3.3

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 4.2.3.3.1

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

$f^{- 1} (- 9 \sqrt{x}) = \frac{81 {(x^{\frac{1}{2}})}^{2}}{81}$

Step 4.2.3.3.2

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

$f^{- 1} (- 9 \sqrt{x}) = \frac{81 x^{\frac{1}{2} \cdot 2}}{81}$

Step 4.2.3.3.3

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

$f^{- 1} (- 9 \sqrt{x}) = \frac{81 x^{\frac{2}{2}}}{81}$

Step 4.2.3.3.4

Cancel the common factor of $2$ .

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Step 4.2.3.3.4.1

Cancel the common factor.

$f^{- 1} (- 9 \sqrt{x}) = \frac{81 x^{\frac{2}{2}}}{81}$

Step 4.2.3.3.4.2

Rewrite the expression.

$f^{- 1} (- 9 \sqrt{x}) = \frac{81 x}{81}$

Step 4.2.3.3.5

Simplify.

$f^{- 1} (- 9 \sqrt{x}) = \frac{81 x}{81}$

Step 4.2.4

Cancel the common factor of $81$ .

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Step 4.2.4.1

Cancel the common factor.

$f^{- 1} (- 9 \sqrt{x}) = \frac{81 x}{81}$

Step 4.2.4.2

Divide $x$ by $1$ .

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

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (\frac{x^{2}}{81})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{2}}{81}) = - 9 \sqrt{\frac{x^{2}}{81}}$

Step 4.3.3

Remove parentheses.

$f (\frac{x^{2}}{81}) = - 9 \sqrt{\frac{x^{2}}{81}}$

Step 4.3.4

Rewrite $81$ as $9^{2}$ .

$f (\frac{x^{2}}{81}) = - 9 \sqrt{\frac{x^{2}}{9^{2}}}$

Step 4.3.5

Rewrite $\frac{x^{2}}{9^{2}}$ as ${(\frac{x}{9})}^{2}$ .

$f (\frac{x^{2}}{81}) = - 9 \sqrt{{(\frac{x}{9})}^{2}}$

Step 4.3.6

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

$f (\frac{x^{2}}{81}) = - 9 \frac{x}{9}$

Step 4.3.7

Cancel the common factor of $9$ .

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Step 4.3.7.1

Factor $9$ out of $- 9$ .

$f (\frac{x^{2}}{81}) = 9 (- 1) (\frac{x}{9})$

Step 4.3.7.2

Cancel the common factor.

$f (\frac{x^{2}}{81}) = 9 \cdot (- 1 \frac{x}{9})$

Step 4.3.7.3

Rewrite the expression.

$f (\frac{x^{2}}{81}) = - 1 x$

Step 4.3.8

Rewrite $- 1 x$ as $- x$ .

$f (\frac{x^{2}}{81}) = - x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{2}}{81}$ is the inverse of $f (x) = - 9 \sqrt{x}$ .

$f^{- 1} (x) = \frac{x^{2}}{81}$