Precalculus Examples

$f (x) = \frac{6}{\sqrt{x}}$

Step 1

Write $f (x) = \frac{6}{\sqrt{x}}$ as an equation.

$y = \frac{6}{\sqrt{x}}$

Step 2

Interchange the variables.

$x = \frac{6}{\sqrt{y}}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{6}{\sqrt{y}} = x$ .

$\frac{6}{\sqrt{y}} = x$

Step 3.2

Cross multiply.

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Step 3.2.1

Cross multiply by setting the product of the numerator of the right side and the denominator of the left side equal to the product of the numerator of the left side and the denominator of the right side.

$x \cdot (\sqrt{y}) = 6$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Multiply $x$ by $\sqrt{y}$ .

$x \sqrt{y} = 6$

Step 3.3

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

${(x \sqrt{y})}^{2} = 6^{2}$

Step 3.4

Simplify each side of the equation.

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Step 3.4.1

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

${(x y^{\frac{1}{2}})}^{2} = 6^{2}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

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

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Step 3.4.2.1.1

Apply the product rule to $x y^{\frac{1}{2}}$ .

$x^{2} {(y^{\frac{1}{2}})}^{2} = 6^{2}$

Step 3.4.2.1.2

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

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Step 3.4.2.1.2.1

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

$x^{2} y^{\frac{1}{2} \cdot 2} = 6^{2}$

Step 3.4.2.1.2.2

Cancel the common factor of $2$ .

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Step 3.4.2.1.2.2.1

Cancel the common factor.

$x^{2} y^{\frac{1}{2} \cdot 2} = 6^{2}$

Step 3.4.2.1.2.2.2

Rewrite the expression.

$x^{2} y^{1} = 6^{2}$

Step 3.4.2.1.3

Simplify.

$x^{2} y = 6^{2}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Raise $6$ to the power of $2$ .

$x^{2} y = 36$

Step 3.5

Divide each term in $x^{2} y = 36$ by $x^{2}$ and simplify.

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Step 3.5.1

Divide each term in $x^{2} y = 36$ by $x^{2}$ .

$\frac{x^{2} y}{x^{2}} = \frac{36}{x^{2}}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $x^{2}$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{x^{2} y}{x^{2}} = \frac{36}{x^{2}}$

Step 3.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{36}{x^{2}}$

Step 4

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

$f^{- 1} (x) = \frac{36}{x^{2}}$

Step 5

Verify if $f^{- 1} (x) = \frac{36}{x^{2}}$ is the inverse of $f (x) = \frac{6}{\sqrt{x}}$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (\frac{6}{\sqrt{x}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{6}{\sqrt{x}}) = \frac{36}{{(\frac{6}{\sqrt{x}})}^{2}}$

Step 5.2.3

Simplify the denominator.

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Step 5.2.3.1

Apply the product rule to $\frac{6}{\sqrt{x}}$ .

$f^{- 1} (\frac{6}{\sqrt{x}}) = \frac{36}{\frac{6^{2}}{{\sqrt{x}}^{2}}}$

Step 5.2.3.2

Raise $6$ to the power of $2$ .

$f^{- 1} (\frac{6}{\sqrt{x}}) = \frac{36}{\frac{36}{{\sqrt{x}}^{2}}}$

Step 5.2.3.3

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

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Step 5.2.3.3.1

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

$f^{- 1} (\frac{6}{\sqrt{x}}) = \frac{36}{\frac{36}{{(x^{\frac{1}{2}})}^{2}}}$

Step 5.2.3.3.2

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

$f^{- 1} (\frac{6}{\sqrt{x}}) = \frac{36}{\frac{36}{x^{\frac{1}{2} \cdot 2}}}$

Step 5.2.3.3.3

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

$f^{- 1} (\frac{6}{\sqrt{x}}) = \frac{36}{\frac{36}{x^{\frac{2}{2}}}}$

Step 5.2.3.3.4

Cancel the common factor of $2$ .

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Step 5.2.3.3.4.1

Cancel the common factor.

$f^{- 1} (\frac{6}{\sqrt{x}}) = \frac{36}{\frac{36}{x^{\frac{2}{2}}}}$

Step 5.2.3.3.4.2

Rewrite the expression.

$f^{- 1} (\frac{6}{\sqrt{x}}) = \frac{36}{\frac{36}{x}}$

Step 5.2.3.3.5

Simplify.

$f^{- 1} (\frac{6}{\sqrt{x}}) = \frac{36}{\frac{36}{x}}$

Step 5.2.4

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{6}{\sqrt{x}}) = 36 (\frac{x}{36})$

Step 5.2.5

Cancel the common factor of $36$ .

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Step 5.2.5.1

Cancel the common factor.

$f^{- 1} (\frac{6}{\sqrt{x}}) = 36 (\frac{x}{36})$

Step 5.2.5.2

Rewrite the expression.

$f^{- 1} (\frac{6}{\sqrt{x}}) = x$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (\frac{36}{x^{2}})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{36}{x^{2}}) = \frac{6}{\sqrt{\frac{36}{x^{2}}}}$

Step 5.3.3

Remove parentheses.

$f (\frac{36}{x^{2}}) = \frac{6}{\sqrt{\frac{36}{x^{2}}}}$

Step 5.3.4

Simplify the denominator.

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Step 5.3.4.1

Rewrite $36$ as $6^{2}$ .

$f (\frac{36}{x^{2}}) = \frac{6}{\sqrt{\frac{6^{2}}{x^{2}}}}$

Step 5.3.4.2

Rewrite $\frac{6^{2}}{x^{2}}$ as ${(\frac{6}{x})}^{2}$ .

$f (\frac{36}{x^{2}}) = \frac{6}{\sqrt{{(\frac{6}{x})}^{2}}}$

Step 5.3.4.3

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

$f (\frac{36}{x^{2}}) = \frac{6}{\frac{6}{x}}$

Step 5.3.5

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{36}{x^{2}}) = 6 (\frac{x}{6})$

Step 5.3.6

Cancel the common factor of $6$ .

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Step 5.3.6.1

Cancel the common factor.

$f (\frac{36}{x^{2}}) = 6 (\frac{x}{6})$

Step 5.3.6.2

Rewrite the expression.

$f (\frac{36}{x^{2}}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{36}{x^{2}}$ is the inverse of $f (x) = \frac{6}{\sqrt{x}}$ .

$f^{- 1} (x) = \frac{36}{x^{2}}$