Calculus Examples

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

Step 1

Write $f (x) = \sqrt{1 - 9 x}$ as an equation.

$y = \sqrt{1 - 9 x}$

Step 2

Interchange the variables.

$x = \sqrt{1 - 9 y}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\sqrt{1 - 9 y} = x$ .

$\sqrt{1 - 9 y} = x$

Step 3.2

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

${\sqrt{1 - 9 y}}^{2} = x^{2}$

Step 3.3

Simplify each side of the equation.

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Step 3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 - 9 y}$ as ${(1 - 9 y)}^{\frac{1}{2}}$ .

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Simplify ${({(1 - 9 y)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1

Multiply the exponents in ${({(1 - 9 y)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1.1

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

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

Step 3.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.2.1.1.2.1

Cancel the common factor.

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

Step 3.3.2.1.1.2.2

Rewrite the expression.

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

Step 3.3.2.1.2

Simplify.

$1 - 9 y = x^{2}$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $1$ from both sides of the equation.

$- 9 y = x^{2} - 1$

Step 3.4.2

Divide each term in $- 9 y = x^{2} - 1$ by $- 9$ and simplify.

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Step 3.4.2.1

Divide each term in $- 9 y = x^{2} - 1$ by $- 9$ .

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

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $- 9$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

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

Step 3.4.2.2.1.2

Divide $y$ by $1$ .

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

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Simplify each term.

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Step 3.4.2.3.1.1

Move the negative in front of the fraction.

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

Step 3.4.2.3.1.2

Dividing two negative values results in a positive value.

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Combine the numerators over the common denominator.

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

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

Rewrite ${\sqrt{1 - 9 x}}^{2}$ as $1 - 9 x$ .

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Step 5.2.4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 - 9 x}$ as ${(1 - 9 x)}^{\frac{1}{2}}$ .

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

Step 5.2.4.1.2

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

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

Step 5.2.4.1.3

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

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

Step 5.2.4.1.4

Cancel the common factor of $2$ .

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Step 5.2.4.1.4.1

Cancel the common factor.

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

Step 5.2.4.1.4.2

Rewrite the expression.

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

Step 5.2.4.1.5

Simplify.

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

Step 5.2.4.2

Apply the distributive property.

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

Step 5.2.4.3

Multiply $- 1$ by $1$ .

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

Step 5.2.4.4

Multiply $- 9$ by $- 1$ .

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

Step 5.2.5

Simplify terms.

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Step 5.2.5.1

Combine the opposite terms in $- 1 + 9 x + 1$ .

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Step 5.2.5.1.1

Add $- 1$ and $1$ .

$f^{- 1} (\sqrt{1 - 9 x}) = \frac{9 x + 0}{9}$

Step 5.2.5.1.2

Add $9 x$ and $0$ .

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

Step 5.2.5.2

Cancel the common factor of $9$ .

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Step 5.2.5.2.1

Cancel the common factor.

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

Step 5.2.5.2.2

Divide $x$ by $1$ .

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

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 $f (- \frac{x^{2}}{9} + \frac{1}{9})$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Step 5.3.3

Apply the distributive property.

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

Step 5.3.4

Cancel the common factor of $9$ .

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Step 5.3.4.1

Move the leading negative in $- \frac{x^{2}}{9}$ into the numerator.

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

Step 5.3.4.2

Factor $9$ out of $- 9$ .

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

Step 5.3.4.3

Cancel the common factor.

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

Step 5.3.4.4

Rewrite the expression.

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

Step 5.3.5

Multiply.

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Step 5.3.5.1

Multiply $- 1$ by $- 1$ .

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

Step 5.3.5.2

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

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

Step 5.3.6

Cancel the common factor of $9$ .

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Step 5.3.6.1

Factor $9$ out of $- 9$ .

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

Step 5.3.6.2

Cancel the common factor.

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

Step 5.3.6.3

Rewrite the expression.

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

Step 5.3.7

Simplify by subtracting numbers.

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Step 5.3.7.1

Subtract $1$ from $1$ .

$f (- \frac{x^{2}}{9} + \frac{1}{9}) = \sqrt{x^{2} + 0}$

Step 5.3.7.2

Add $x^{2}$ and $0$ .

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

Step 5.3.8

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

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

Step 5.4

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

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