Precalculus Examples

$f (x) = \sqrt{8 - x}$

Step 1

Write $f (x) = \sqrt{8 - x}$ as an equation.

$y = \sqrt{8 - x}$

Step 2

Interchange the variables.

$x = \sqrt{8 - y}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\sqrt{8 - y} = x$ .

$\sqrt{8 - y} = x$

Step 3.2

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

${\sqrt{8 - y}}^{2} = x^{2}$

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 $\sqrt{8 - y}$ as ${(8 - y)}^{\frac{1}{2}}$ .

${({(8 - y)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Simplify ${({(8 - y)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1

Multiply the exponents in ${({(8 - y)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1.1

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

${(8 - y)}^{\frac{1}{2} \cdot 2} = x^{2}$

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.

${(8 - y)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 3.3.2.1.1.2.2

Rewrite the expression.

${(8 - y)}^{1} = x^{2}$

Step 3.3.2.1.2

Simplify.

$8 - y = x^{2}$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $8$ from both sides of the equation.

$- y = x^{2} - 8$

Step 3.4.2

Divide each term in $- y = x^{2} - 8$ by $- 1$ and simplify.

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Step 3.4.2.1

Divide each term in $- y = x^{2} - 8$ by $- 1$ .

$\frac{- y}{- 1} = \frac{x^{2}}{- 1} + \frac{- 8}{- 1}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{x^{2}}{- 1} + \frac{- 8}{- 1}$

Step 3.4.2.2.2

Divide $y$ by $1$ .

$y = \frac{x^{2}}{- 1} + \frac{- 8}{- 1}$

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 one from the denominator of $\frac{x^{2}}{- 1}$ .

$y = - 1 \cdot x^{2} + \frac{- 8}{- 1}$

Step 3.4.2.3.1.2

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

$y = - x^{2} + \frac{- 8}{- 1}$

Step 3.4.2.3.1.3

Divide $- 8$ by $- 1$ .

$y = - x^{2} + 8$

Step 4

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

$f^{- 1} (x) = - x^{2} + 8$

Step 5

Verify if $f^{- 1} (x) = - x^{2} + 8$ is the inverse of $f (x) = \sqrt{8 - x}$ .

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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 $f^{- 1} (\sqrt{8 - x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt{8 - x}) = - {(\sqrt{8 - x})}^{2} + 8$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Rewrite ${\sqrt{8 - x}}^{2}$ as $8 - x$ .

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Step 5.2.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8 - x}$ as ${(8 - x)}^{\frac{1}{2}}$ .

$f^{- 1} (\sqrt{8 - x}) = - {({(8 - x)}^{\frac{1}{2}})}^{2} + 8$

Step 5.2.3.1.2

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

$f^{- 1} (\sqrt{8 - x}) = - {(8 - x)}^{\frac{1}{2} \cdot 2} + 8$

Step 5.2.3.1.3

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

$f^{- 1} (\sqrt{8 - x}) = - {(8 - x)}^{\frac{2}{2}} + 8$

Step 5.2.3.1.4

Cancel the common factor of $2$ .

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Step 5.2.3.1.4.1

Cancel the common factor.

$f^{- 1} (\sqrt{8 - x}) = - {(8 - x)}^{\frac{2}{2}} + 8$

Step 5.2.3.1.4.2

Rewrite the expression.

$f^{- 1} (\sqrt{8 - x}) = - (8 - x) + 8$

Step 5.2.3.1.5

Simplify.

$f^{- 1} (\sqrt{8 - x}) = - (8 - x) + 8$

Step 5.2.3.2

Apply the distributive property.

$f^{- 1} (\sqrt{8 - x}) = - 1 \cdot 8 + x + 8$

Step 5.2.3.3

Multiply $- 1$ by $8$ .

$f^{- 1} (\sqrt{8 - x}) = - 8 + x + 8$

Step 5.2.3.4

Multiply $- - x$ .

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Step 5.2.3.4.1

Multiply $- 1$ by $- 1$ .

$f^{- 1} (\sqrt{8 - x}) = - 8 + 1 x + 8$

Step 5.2.3.4.2

Multiply $x$ by $1$ .

$f^{- 1} (\sqrt{8 - x}) = - 8 + x + 8$

Step 5.2.4

Combine the opposite terms in $- 8 + x + 8$ .

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Step 5.2.4.1

Add $- 8$ and $8$ .

$f^{- 1} (\sqrt{8 - x}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} (\sqrt{8 - x}) = x$

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 $f (- x^{2} + 8)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- x^{2} + 8) = \sqrt{8 - (- x^{2} + 8)}$

Step 5.3.3

Apply the distributive property.

$f (- x^{2} + 8) = \sqrt{8 + x^{2} - 1 \cdot 8}$

Step 5.3.4

Multiply $- 1$ by $8$ .

$f (- x^{2} + 8) = \sqrt{8 + x^{2} - 8}$

Step 5.3.5

Subtract $8$ from $8$ .

$f (- x^{2} + 8) = \sqrt{x^{2} + 0}$

Step 5.3.6

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

$f (- x^{2} + 8) = \sqrt{x^{2}}$

Step 5.3.7

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

$f (- x^{2} + 8) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - x^{2} + 8$ is the inverse of $f (x) = \sqrt{8 - x}$ .

$f^{- 1} (x) = - x^{2} + 8$