Calculus Examples

$\sqrt{\frac{1}{x - 2}}$

Step 1

Interchange the variables.

$x = \sqrt{\frac{1}{y - 2}}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\sqrt{\frac{1}{y - 2}} = x$ .

$\sqrt{\frac{1}{y - 2}} = x$

Step 2.2

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

${\sqrt{\frac{1}{y - 2}}}^{2} = x^{2}$

Step 2.3

Simplify each side of the equation.

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Step 2.3.1

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

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

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

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Step 2.3.2.1.1

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

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Step 2.3.2.1.1.1

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

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

Step 2.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.3.2.1.1.2.1

Cancel the common factor.

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

Step 2.3.2.1.1.2.2

Rewrite the expression.

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

Step 2.3.2.1.2

Simplify.

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

Step 2.4

Solve for $y$ .

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Step 2.4.1

Find the LCD of the terms in the equation.

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Step 2.4.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y - 2, 1$

Step 2.4.1.2

Remove parentheses.

$y - 2, 1$

Step 2.4.1.3

The LCM of one and any expression is the expression.

$y - 2$

Step 2.4.2

Multiply each term in $\frac{1}{y - 2} = x^{2}$ by $y - 2$ to eliminate the fractions.

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Step 2.4.2.1

Multiply each term in $\frac{1}{y - 2} = x^{2}$ by $y - 2$ .

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

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $y - 2$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

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

Step 2.4.2.2.1.2

Rewrite the expression.

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

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

Apply the distributive property.

$1 = x^{2} y + x^{2} \cdot - 2$

Step 2.4.2.3.2

Move $- 2$ to the left of $x^{2}$ .

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

Step 2.4.3

Solve the equation.

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Step 2.4.3.1

Rewrite the equation as $x^{2} y - 2 x^{2} = 1$ .

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

Step 2.4.3.2

Add $2 x^{2}$ to both sides of the equation.

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

Step 2.4.3.3

Divide each term in $x^{2} y = 1 + 2 x^{2}$ by $x^{2}$ and simplify.

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Step 2.4.3.3.1

Divide each term in $x^{2} y = 1 + 2 x^{2}$ by $x^{2}$ .

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

Step 2.4.3.3.2

Simplify the left side.

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Step 2.4.3.3.2.1

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

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Step 2.4.3.3.2.1.1

Cancel the common factor.

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

Step 2.4.3.3.2.1.2

Divide $y$ by $1$ .

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

Step 2.4.3.3.3

Simplify the right side.

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Step 2.4.3.3.3.1

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

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Step 2.4.3.3.3.1.1

Cancel the common factor.

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

Step 2.4.3.3.3.1.2

Divide $2$ by $1$ .

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

Step 3

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

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

Step 4

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

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

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

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Simplify the denominator.

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Step 4.2.3.1.1

Rewrite $\sqrt{\frac{1}{x - 2}}$ as $\frac{\sqrt{1}}{\sqrt{x - 2}}$ .

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

Step 4.2.3.1.2

Any root of $1$ is $1$ .

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

Step 4.2.3.1.3

Multiply $\frac{1}{\sqrt{x - 2}}$ by $\frac{\sqrt{x - 2}}{\sqrt{x - 2}}$ .

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

Step 4.2.3.1.4

Combine and simplify the denominator.

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Step 4.2.3.1.4.1

Multiply $\frac{1}{\sqrt{x - 2}}$ by $\frac{\sqrt{x - 2}}{\sqrt{x - 2}}$ .

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

Step 4.2.3.1.4.2

Raise $\sqrt{x - 2}$ to the power of $1$ .

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

Step 4.2.3.1.4.3

Raise $\sqrt{x - 2}$ to the power of $1$ .

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

Step 4.2.3.1.4.4

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

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

Step 4.2.3.1.4.5

Add $1$ and $1$ .

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

Step 4.2.3.1.4.6

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

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Step 4.2.3.1.4.6.1

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

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

Step 4.2.3.1.4.6.2

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

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

Step 4.2.3.1.4.6.3

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

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

Step 4.2.3.1.4.6.4

Cancel the common factor of $2$ .

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Step 4.2.3.1.4.6.4.1

Cancel the common factor.

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

Step 4.2.3.1.4.6.4.2

Rewrite the expression.

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

Step 4.2.3.1.4.6.5

Simplify.

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

Step 4.2.3.1.5

Apply the product rule to $\frac{\sqrt{x - 2}}{x - 2}$ .

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

Step 4.2.3.1.6

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

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Step 4.2.3.1.6.1

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

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

Step 4.2.3.1.6.2

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

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

Step 4.2.3.1.6.3

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

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

Step 4.2.3.1.6.4

Cancel the common factor of $2$ .

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Step 4.2.3.1.6.4.1

Cancel the common factor.

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

Step 4.2.3.1.6.4.2

Rewrite the expression.

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

Step 4.2.3.1.6.5

Simplify.

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

Step 4.2.3.1.7

Cancel the common factor of $x - 2$ and ${(x - 2)}^{2}$ .

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Step 4.2.3.1.7.1

Multiply by $1$ .

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

Step 4.2.3.1.7.2

Cancel the common factors.

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Step 4.2.3.1.7.2.1

Factor $x - 2$ out of ${(x - 2)}^{2}$ .

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

Step 4.2.3.1.7.2.2

Cancel the common factor.

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

Step 4.2.3.1.7.2.3

Rewrite the expression.

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

Step 4.2.3.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2.3.3

Multiply $x - 2$ by $1$ .

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

Step 4.2.4

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

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Step 4.2.4.1

Add $- 2$ and $2$ .

$f^{- 1} (\sqrt{\frac{1}{x - 2}}) = x + 0$

Step 4.2.4.2

Add $x$ and $0$ .

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

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

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

Step 4.3.3

Subtract $2$ from $2$ .

$f (\frac{1}{x^{2}} + 2) = \sqrt{\frac{1}{\frac{1}{x^{2}} + 0}}$

Step 4.3.4

Add $\frac{1}{x^{2}}$ and $0$ .

$f (\frac{1}{x^{2}} + 2) = \sqrt{\frac{1}{\frac{1}{x^{2}}}}$

Step 4.3.5

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{1}{x^{2}} + 2) = \sqrt{1 x^{2}}$

Step 4.3.6

Multiply $x^{2}$ by $1$ .

$f (\frac{1}{x^{2}} + 2) = \sqrt{x^{2}}$

Step 4.3.7

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

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

Step 4.4

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

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