Precalculus Examples

$f (x) = 4 + \sqrt{x + 4}$

Step 1

Write $f (x) = 4 + \sqrt{x + 4}$ as an equation.

$y = 4 + \sqrt{x + 4}$

Step 2

Interchange the variables.

$x = 4 + \sqrt{y + 4}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $4 + \sqrt{y + 4} = x$ .

$4 + \sqrt{y + 4} = x$

Step 3.2

Subtract $4$ from both sides of the equation.

$\sqrt{y + 4} = x - 4$

Step 3.3

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

${\sqrt{y + 4}}^{2} = {(x - 4)}^{2}$

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 $\sqrt{y + 4}$ as ${(y + 4)}^{\frac{1}{2}}$ .

${({(y + 4)}^{\frac{1}{2}})}^{2} = {(x - 4)}^{2}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Simplify ${({(y + 4)}^{\frac{1}{2}})}^{2}$ .

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Step 3.4.2.1.1

Multiply the exponents in ${({(y + 4)}^{\frac{1}{2}})}^{2}$ .

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Step 3.4.2.1.1.1

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

${(y + 4)}^{\frac{1}{2} \cdot 2} = {(x - 4)}^{2}$

Step 3.4.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.4.2.1.1.2.1

Cancel the common factor.

${(y + 4)}^{\frac{1}{2} \cdot 2} = {(x - 4)}^{2}$

Step 3.4.2.1.1.2.2

Rewrite the expression.

${(y + 4)}^{1} = {(x - 4)}^{2}$

Step 3.4.2.1.2

Simplify.

$y + 4 = {(x - 4)}^{2}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify ${(x - 4)}^{2}$ .

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Step 3.4.3.1.1

Rewrite ${(x - 4)}^{2}$ as $(x - 4) (x - 4)$ .

$y + 4 = (x - 4) (x - 4)$

Step 3.4.3.1.2

Expand $(x - 4) (x - 4)$ using the FOIL Method.

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Step 3.4.3.1.2.1

Apply the distributive property.

$y + 4 = x (x - 4) - 4 (x - 4)$

Step 3.4.3.1.2.2

Apply the distributive property.

$y + 4 = x \cdot x + x \cdot - 4 - 4 (x - 4)$

Step 3.4.3.1.2.3

Apply the distributive property.

$y + 4 = x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4$

Step 3.4.3.1.3

Simplify and combine like terms.

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Step 3.4.3.1.3.1

Simplify each term.

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Step 3.4.3.1.3.1.1

Multiply $x$ by $x$ .

$y + 4 = x^{2} + x \cdot - 4 - 4 x - 4 \cdot - 4$

Step 3.4.3.1.3.1.2

Move $- 4$ to the left of $x$ .

$y + 4 = x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4$

Step 3.4.3.1.3.1.3

Multiply $- 4$ by $- 4$ .

$y + 4 = x^{2} - 4 x - 4 x + 16$

Step 3.4.3.1.3.2

Subtract $4 x$ from $- 4 x$ .

$y + 4 = x^{2} - 8 x + 16$

Step 3.5

Move all terms not containing $y$ to the right side of the equation.

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Step 3.5.1

Subtract $4$ from both sides of the equation.

$y = x^{2} - 8 x + 16 - 4$

Step 3.5.2

Subtract $4$ from $16$ .

$y = x^{2} - 8 x + 12$

Step 4

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

$f^{- 1} (x) = x^{2} - 8 x + 12$

Step 5

Verify if $f^{- 1} (x) = x^{2} - 8 x + 12$ is the inverse of $f (x) = 4 + \sqrt{x + 4}$ .

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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 $f^{- 1} (4 + \sqrt{x + 4})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (4 + \sqrt{x + 4}) = {(4 + \sqrt{x + 4})}^{2} - 8 (4 + \sqrt{x + 4}) + 12$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Rewrite ${(4 + \sqrt{x + 4})}^{2}$ as $(4 + \sqrt{x + 4}) (4 + \sqrt{x + 4})$ .

$f^{- 1} (4 + \sqrt{x + 4}) = (4 + \sqrt{x + 4}) (4 + \sqrt{x + 4}) - 8 (4 + \sqrt{x + 4}) + 12$

Step 5.2.3.2

Expand $(4 + \sqrt{x + 4}) (4 + \sqrt{x + 4})$ using the FOIL Method.

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Step 5.2.3.2.1

Apply the distributive property.

$f^{- 1} (4 + \sqrt{x + 4}) = 4 (4 + \sqrt{x + 4}) + \sqrt{x + 4} (4 + \sqrt{x + 4}) - 8 (4 + \sqrt{x + 4}) + 12$

Step 5.2.3.2.2

Apply the distributive property.

$f^{- 1} (4 + \sqrt{x + 4}) = 4 \cdot 4 + 4 \sqrt{x + 4} + \sqrt{x + 4} (4 + \sqrt{x + 4}) - 8 (4 + \sqrt{x + 4}) + 12$

Step 5.2.3.2.3

Apply the distributive property.

$f^{- 1} (4 + \sqrt{x + 4}) = 4 \cdot 4 + 4 \sqrt{x + 4} + \sqrt{x + 4} \cdot 4 + \sqrt{x + 4} \sqrt{x + 4} - 8 (4 + \sqrt{x + 4}) + 12$

Step 5.2.3.3

Simplify and combine like terms.

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Step 5.2.3.3.1

Simplify each term.

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Step 5.2.3.3.1.1

Multiply $4$ by $4$ .

$f^{- 1} (4 + \sqrt{x + 4}) = 16 + 4 \sqrt{x + 4} + \sqrt{x + 4} \cdot 4 + \sqrt{x + 4} \sqrt{x + 4} - 8 (4 + \sqrt{x + 4}) + 12$

Step 5.2.3.3.1.2

Move $4$ to the left of $\sqrt{x + 4}$ .

$f^{- 1} (4 + \sqrt{x + 4}) = 16 + 4 \sqrt{x + 4} + 4 \sqrt{x + 4} + \sqrt{x + 4} \sqrt{x + 4} - 8 (4 + \sqrt{x + 4}) + 12$

Step 5.2.3.3.1.3

Multiply $\sqrt{x + 4} \sqrt{x + 4}$ .

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Step 5.2.3.3.1.3.1

Raise $\sqrt{x + 4}$ to the power of $1$ .

$f^{- 1} (4 + \sqrt{x + 4}) = 16 + 4 \sqrt{x + 4} + 4 \sqrt{x + 4} + \sqrt{x + 4} \sqrt{x + 4} - 8 (4 + \sqrt{x + 4}) + 12$

Step 5.2.3.3.1.3.2

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

$f^{- 1} (4 + \sqrt{x + 4}) = 16 + 4 \sqrt{x + 4} + 4 \sqrt{x + 4} + {\sqrt{x + 4}}^{1 + 1} - 8 (4 + \sqrt{x + 4}) + 12$

Step 5.2.3.3.1.3.3

Add $1$ and $1$ .

$f^{- 1} (4 + \sqrt{x + 4}) = 16 + 4 \sqrt{x + 4} + 4 \sqrt{x + 4} + {\sqrt{x + 4}}^{2} - 8 (4 + \sqrt{x + 4}) + 12$

Step 5.2.3.3.1.4

Rewrite ${\sqrt{x + 4}}^{2}$ as $x + 4$ .

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Step 5.2.3.3.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 4}$ as ${(x + 4)}^{\frac{1}{2}}$ .

$f^{- 1} (4 + \sqrt{x + 4}) = 16 + 4 \sqrt{x + 4} + 4 \sqrt{x + 4} + {({(x + 4)}^{\frac{1}{2}})}^{2} - 8 (4 + \sqrt{x + 4}) + 12$

Step 5.2.3.3.1.4.2

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

$f^{- 1} (4 + \sqrt{x + 4}) = 16 + 4 \sqrt{x + 4} + 4 \sqrt{x + 4} + {(x + 4)}^{\frac{1}{2} \cdot 2} - 8 (4 + \sqrt{x + 4}) + 12$

Step 5.2.3.3.1.4.3

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

$f^{- 1} (4 + \sqrt{x + 4}) = 16 + 4 \sqrt{x + 4} + 4 \sqrt{x + 4} + {(x + 4)}^{\frac{2}{2}} - 8 (4 + \sqrt{x + 4}) + 12$

Step 5.2.3.3.1.4.4

Cancel the common factor of $2$ .

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Step 5.2.3.3.1.4.4.1

Cancel the common factor.

$f^{- 1} (4 + \sqrt{x + 4}) = 16 + 4 \sqrt{x + 4} + 4 \sqrt{x + 4} + {(x + 4)}^{\frac{2}{2}} - 8 (4 + \sqrt{x + 4}) + 12$

Step 5.2.3.3.1.4.4.2

Rewrite the expression.

$f^{- 1} (4 + \sqrt{x + 4}) = 16 + 4 \sqrt{x + 4} + 4 \sqrt{x + 4} + x + 4 - 8 (4 + \sqrt{x + 4}) + 12$

Step 5.2.3.3.1.4.5

Simplify.

$f^{- 1} (4 + \sqrt{x + 4}) = 16 + 4 \sqrt{x + 4} + 4 \sqrt{x + 4} + x + 4 - 8 (4 + \sqrt{x + 4}) + 12$

Step 5.2.3.3.2

Add $16$ and $4$ .

$f^{- 1} (4 + \sqrt{x + 4}) = 20 + 4 \sqrt{x + 4} + 4 \sqrt{x + 4} + x - 8 (4 + \sqrt{x + 4}) + 12$

Step 5.2.3.3.3

Add $4 \sqrt{x + 4}$ and $4 \sqrt{x + 4}$ .

$f^{- 1} (4 + \sqrt{x + 4}) = 20 + 8 \sqrt{x + 4} + x - 8 (4 + \sqrt{x + 4}) + 12$

Step 5.2.3.4

Apply the distributive property.

$f^{- 1} (4 + \sqrt{x + 4}) = 20 + 8 \sqrt{x + 4} + x - 8 \cdot 4 - 8 \sqrt{x + 4} + 12$

Step 5.2.3.5

Multiply $- 8$ by $4$ .

$f^{- 1} (4 + \sqrt{x + 4}) = 20 + 8 \sqrt{x + 4} + x - 32 - 8 \sqrt{x + 4} + 12$

Step 5.2.4

Simplify by adding terms.

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Step 5.2.4.1

Combine the opposite terms in $20 + 8 \sqrt{x + 4} + x - 32 - 8 \sqrt{x + 4} + 12$ .

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Step 5.2.4.1.1

Subtract $8 \sqrt{x + 4}$ from $8 \sqrt{x + 4}$ .

$f^{- 1} (4 + \sqrt{x + 4}) = 20 + x - 32 + 0 + 12$

Step 5.2.4.1.2

Add $20 + x - 32$ and $0$ .

$f^{- 1} (4 + \sqrt{x + 4}) = 20 + x - 32 + 12$

Step 5.2.4.2

Subtract $32$ from $20$ .

$f^{- 1} (4 + \sqrt{x + 4}) = x - 12 + 12$

Step 5.2.4.3

Combine the opposite terms in $x - 12 + 12$ .

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Step 5.2.4.3.1

Add $- 12$ and $12$ .

$f^{- 1} (4 + \sqrt{x + 4}) = x + 0$

Step 5.2.4.3.2

Add $x$ and $0$ .

$f^{- 1} (4 + \sqrt{x + 4}) = x$

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 $f (x^{2} - 8 x + 12)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x^{2} - 8 x + 12) = 4 + \sqrt{(x^{2} - 8 x + 12) + 4}$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Add $12$ and $4$ .

$f (x^{2} - 8 x + 12) = 4 + \sqrt{x^{2} - 8 x + 16}$

Step 5.3.3.2

Factor using the perfect square rule.

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Step 5.3.3.2.1

Rewrite $16$ as $4^{2}$ .

$f (x^{2} - 8 x + 12) = 4 + \sqrt{x^{2} - 8 x + 4^{2}}$

Step 5.3.3.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 x = 2 \cdot x \cdot 4$

Step 5.3.3.2.3

Rewrite the polynomial.

$f (x^{2} - 8 x + 12) = 4 + \sqrt{x^{2} - 2 \cdot x \cdot 4 + 4^{2}}$

Step 5.3.3.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 4$ .

$f (x^{2} - 8 x + 12) = 4 + \sqrt{{(x - 4)}^{2}}$

Step 5.3.3.3

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

$f (x^{2} - 8 x + 12) = 4 + x - 4$

Step 5.3.4

Combine the opposite terms in $4 + x - 4$ .

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Step 5.3.4.1

Subtract $4$ from $4$ .

$f (x^{2} - 8 x + 12) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (x^{2} - 8 x + 12) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x^{2} - 8 x + 12$ is the inverse of $f (x) = 4 + \sqrt{x + 4}$ .

$f^{- 1} (x) = x^{2} - 8 x + 12$