Precalculus Examples

$f (x) = \sqrt{x} + 10$ ; find $f^{- 1} (x)$

Step 1

Write $f (x) = \sqrt{x} + 10$ as an equation.

$y = \sqrt{x} + 10$

Step 2

Interchange the variables.

$x = \sqrt{y} + 10$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\sqrt{y} + 10 = x$ .

$\sqrt{y} + 10 = x$

Step 3.2

Subtract $10$ from both sides of the equation.

$\sqrt{y} = x - 10$

Step 3.3

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

${\sqrt{y}}^{2} = {(x - 10)}^{2}$

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 $y^{\frac{1}{2}}$ .

${(y^{\frac{1}{2}})}^{2} = {(x - 10)}^{2}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

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

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Step 3.4.2.1.1

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

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Step 3.4.2.1.1.1

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

$y^{\frac{1}{2} \cdot 2} = {(x - 10)}^{2}$

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.

$y^{\frac{1}{2} \cdot 2} = {(x - 10)}^{2}$

Step 3.4.2.1.1.2.2

Rewrite the expression.

$y^{1} = {(x - 10)}^{2}$

Step 3.4.2.1.2

Simplify.

$y = {(x - 10)}^{2}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify ${(x - 10)}^{2}$ .

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Step 3.4.3.1.1

Rewrite ${(x - 10)}^{2}$ as $(x - 10) (x - 10)$ .

$y = (x - 10) (x - 10)$

Step 3.4.3.1.2

Expand $(x - 10) (x - 10)$ using the FOIL Method.

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Step 3.4.3.1.2.1

Apply the distributive property.

$y = x (x - 10) - 10 (x - 10)$

Step 3.4.3.1.2.2

Apply the distributive property.

$y = x \cdot x + x \cdot - 10 - 10 (x - 10)$

Step 3.4.3.1.2.3

Apply the distributive property.

$y = x \cdot x + x \cdot - 10 - 10 x - 10 \cdot - 10$

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 $x$ .

$y = x^{2} + x \cdot - 10 - 10 x - 10 \cdot - 10$

Step 3.4.3.1.3.1.2

Move $- 10$ to the left of $x$ .

$y = x^{2} - 10 \cdot x - 10 x - 10 \cdot - 10$

Step 3.4.3.1.3.1.3

Multiply $- 10$ by $- 10$ .

$y = x^{2} - 10 x - 10 x + 100$

Step 3.4.3.1.3.2

Subtract $10 x$ from $- 10 x$ .

$y = x^{2} - 20 x + 100$

Step 4

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

$f^{- 1} (x) = x^{2} - 20 x + 100$

Step 5

Verify if $f^{- 1} (x) = x^{2} - 20 x + 100$ is the inverse of $f (x) = \sqrt{x} + 10$ .

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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 $f^{- 1} (\sqrt{x} + 10)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt{x} + 10) = {(\sqrt{x} + 10)}^{2} - 20 (\sqrt{x} + 10) + 100$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Rewrite ${(\sqrt{x} + 10)}^{2}$ as $(\sqrt{x} + 10) (\sqrt{x} + 10)$ .

$f^{- 1} (\sqrt{x} + 10) = (\sqrt{x} + 10) (\sqrt{x} + 10) - 20 (\sqrt{x} + 10) + 100$

Step 5.2.3.2

Expand $(\sqrt{x} + 10) (\sqrt{x} + 10)$ using the FOIL Method.

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Step 5.2.3.2.1

Apply the distributive property.

$f^{- 1} (\sqrt{x} + 10) = \sqrt{x} (\sqrt{x} + 10) + 10 (\sqrt{x} + 10) - 20 (\sqrt{x} + 10) + 100$

Step 5.2.3.2.2

Apply the distributive property.

$f^{- 1} (\sqrt{x} + 10) = \sqrt{x} \sqrt{x} + \sqrt{x} \cdot 10 + 10 (\sqrt{x} + 10) - 20 (\sqrt{x} + 10) + 100$

Step 5.2.3.2.3

Apply the distributive property.

$f^{- 1} (\sqrt{x} + 10) = \sqrt{x} \sqrt{x} + \sqrt{x} \cdot 10 + 10 \sqrt{x} + 10 \cdot 10 - 20 (\sqrt{x} + 10) + 100$

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 $\sqrt{x} \sqrt{x}$ .

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Step 5.2.3.3.1.1.1

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

$f^{- 1} (\sqrt{x} + 10) = \sqrt{x} \sqrt{x} + \sqrt{x} \cdot 10 + 10 \sqrt{x} + 10 \cdot 10 - 20 (\sqrt{x} + 10) + 100$

Step 5.2.3.3.1.1.2

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

$f^{- 1} (\sqrt{x} + 10) = \sqrt{x} \sqrt{x} + \sqrt{x} \cdot 10 + 10 \sqrt{x} + 10 \cdot 10 - 20 (\sqrt{x} + 10) + 100$

Step 5.2.3.3.1.1.3

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

$f^{- 1} (\sqrt{x} + 10) = {\sqrt{x}}^{1 + 1} + \sqrt{x} \cdot 10 + 10 \sqrt{x} + 10 \cdot 10 - 20 (\sqrt{x} + 10) + 100$

Step 5.2.3.3.1.1.4

Add $1$ and $1$ .

$f^{- 1} (\sqrt{x} + 10) = {\sqrt{x}}^{2} + \sqrt{x} \cdot 10 + 10 \sqrt{x} + 10 \cdot 10 - 20 (\sqrt{x} + 10) + 100$

Step 5.2.3.3.1.2

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

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Step 5.2.3.3.1.2.1

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

$f^{- 1} (\sqrt{x} + 10) = {(x^{\frac{1}{2}})}^{2} + \sqrt{x} \cdot 10 + 10 \sqrt{x} + 10 \cdot 10 - 20 (\sqrt{x} + 10) + 100$

Step 5.2.3.3.1.2.2

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

$f^{- 1} (\sqrt{x} + 10) = x^{\frac{1}{2} \cdot 2} + \sqrt{x} \cdot 10 + 10 \sqrt{x} + 10 \cdot 10 - 20 (\sqrt{x} + 10) + 100$

Step 5.2.3.3.1.2.3

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

$f^{- 1} (\sqrt{x} + 10) = x^{\frac{2}{2}} + \sqrt{x} \cdot 10 + 10 \sqrt{x} + 10 \cdot 10 - 20 (\sqrt{x} + 10) + 100$

Step 5.2.3.3.1.2.4

Cancel the common factor of $2$ .

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Step 5.2.3.3.1.2.4.1

Cancel the common factor.

$f^{- 1} (\sqrt{x} + 10) = x^{\frac{2}{2}} + \sqrt{x} \cdot 10 + 10 \sqrt{x} + 10 \cdot 10 - 20 (\sqrt{x} + 10) + 100$

Step 5.2.3.3.1.2.4.2

Rewrite the expression.

$f^{- 1} (\sqrt{x} + 10) = x + \sqrt{x} \cdot 10 + 10 \sqrt{x} + 10 \cdot 10 - 20 (\sqrt{x} + 10) + 100$

Step 5.2.3.3.1.2.5

Simplify.

$f^{- 1} (\sqrt{x} + 10) = x + \sqrt{x} \cdot 10 + 10 \sqrt{x} + 10 \cdot 10 - 20 (\sqrt{x} + 10) + 100$

Step 5.2.3.3.1.3

Move $10$ to the left of $\sqrt{x}$ .

$f^{- 1} (\sqrt{x} + 10) = x + 10 \cdot \sqrt{x} + 10 \sqrt{x} + 10 \cdot 10 - 20 (\sqrt{x} + 10) + 100$

Step 5.2.3.3.1.4

Multiply $10$ by $10$ .

$f^{- 1} (\sqrt{x} + 10) = x + 10 \sqrt{x} + 10 \sqrt{x} + 100 - 20 (\sqrt{x} + 10) + 100$

Step 5.2.3.3.2

Add $10 \sqrt{x}$ and $10 \sqrt{x}$ .

$f^{- 1} (\sqrt{x} + 10) = x + 20 \sqrt{x} + 100 - 20 (\sqrt{x} + 10) + 100$

Step 5.2.3.4

Apply the distributive property.

$f^{- 1} (\sqrt{x} + 10) = x + 20 \sqrt{x} + 100 - 20 \sqrt{x} - 20 \cdot 10 + 100$

Step 5.2.3.5

Multiply $- 20$ by $10$ .

$f^{- 1} (\sqrt{x} + 10) = x + 20 \sqrt{x} + 100 - 20 \sqrt{x} - 200 + 100$

Step 5.2.4

Simplify by adding terms.

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Step 5.2.4.1

Combine the opposite terms in $x + 20 \sqrt{x} + 100 - 20 \sqrt{x} - 200 + 100$ .

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Step 5.2.4.1.1

Subtract $20 \sqrt{x}$ from $20 \sqrt{x}$ .

$f^{- 1} (\sqrt{x} + 10) = x + 0 + 100 - 200 + 100$

Step 5.2.4.1.2

Add $x$ and $0$ .

$f^{- 1} (\sqrt{x} + 10) = x + 100 - 200 + 100$

Step 5.2.4.2

Subtract $200$ from $100$ .

$f^{- 1} (\sqrt{x} + 10) = x - 100 + 100$

Step 5.2.4.3

Combine the opposite terms in $x - 100 + 100$ .

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Step 5.2.4.3.1

Add $- 100$ and $100$ .

$f^{- 1} (\sqrt{x} + 10) = x + 0$

Step 5.2.4.3.2

Add $x$ and $0$ .

$f^{- 1} (\sqrt{x} + 10) = x$

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 $f (x^{2} - 20 x + 100)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x^{2} - 20 x + 100) = \sqrt{x^{2} - 20 x + 100} + 10$

Step 5.3.3

Remove parentheses.

$f (x^{2} - 20 x + 100) = \sqrt{x^{2} - 20 x + 100} + 10$

Step 5.3.4

Simplify each term.

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Step 5.3.4.1

Factor using the perfect square rule.

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Step 5.3.4.1.1

Rewrite $100$ as $10^{2}$ .

$f (x^{2} - 20 x + 100) = \sqrt{x^{2} - 20 x + 10^{2}} + 10$

Step 5.3.4.1.2

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

$20 x = 2 \cdot x \cdot 10$

Step 5.3.4.1.3

Rewrite the polynomial.

$f (x^{2} - 20 x + 100) = \sqrt{x^{2} - 2 \cdot x \cdot 10 + 10^{2}} + 10$

Step 5.3.4.1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 10$ .

$f (x^{2} - 20 x + 100) = \sqrt{{(x - 10)}^{2}} + 10$

Step 5.3.4.2

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

$f (x^{2} - 20 x + 100) = x - 10 + 10$

Step 5.3.5

Combine the opposite terms in $x - 10 + 10$ .

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Step 5.3.5.1

Add $- 10$ and $10$ .

$f (x^{2} - 20 x + 100) = x + 0$

Step 5.3.5.2

Add $x$ and $0$ .

$f (x^{2} - 20 x + 100) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x^{2} - 20 x + 100$ is the inverse of $f (x) = \sqrt{x} + 10$ .

$f^{- 1} (x) = x^{2} - 20 x + 100$