Precalculus Examples

$f (x) = x^{\frac{1}{2}} - 7$ ; find $f^{- 1} (x)$

Step 1

Write $f (x) = x^{\frac{1}{2}} - 7$ as an equation.

$y = x^{\frac{1}{2}} - 7$

Step 2

Interchange the variables.

$x = y^{\frac{1}{2}} - 7$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $y^{\frac{1}{2}} - 7 = x$ .

$y^{\frac{1}{2}} - 7 = x$

Step 3.2

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

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Step 3.2.1

Add $7$ to both sides of the equation.

$y^{\frac{1}{2}} - x = 7$

Step 3.2.2

Add $x$ to both sides of the equation.

$y^{\frac{1}{2}} = 7 + x$

Step 3.3

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(y^{\frac{1}{2}})}^{2} = {(7 + x)}^{2}$

Step 3.4

Simplify the exponent.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

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

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Step 3.4.1.1.1

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

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Step 3.4.1.1.1.1

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

$y^{\frac{1}{2} \cdot 2} = {(7 + x)}^{2}$

Step 3.4.1.1.1.2

Cancel the common factor of $2$ .

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Step 3.4.1.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{2} \cdot 2} = {(7 + x)}^{2}$

Step 3.4.1.1.1.2.2

Rewrite the expression.

$y^{1} = {(7 + x)}^{2}$

Step 3.4.1.1.2

Simplify.

$y = {(7 + x)}^{2}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify ${(7 + x)}^{2}$ .

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Step 3.4.2.1.1

Rewrite ${(7 + x)}^{2}$ as $(7 + x) (7 + x)$ .

$y = (7 + x) (7 + x)$

Step 3.4.2.1.2

Expand $(7 + x) (7 + x)$ using the FOIL Method.

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Step 3.4.2.1.2.1

Apply the distributive property.

$y = 7 (7 + x) + x (7 + x)$

Step 3.4.2.1.2.2

Apply the distributive property.

$y = 7 \cdot 7 + 7 x + x (7 + x)$

Step 3.4.2.1.2.3

Apply the distributive property.

$y = 7 \cdot 7 + 7 x + x \cdot 7 + x \cdot x$

Step 3.4.2.1.3

Simplify and combine like terms.

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Step 3.4.2.1.3.1

Simplify each term.

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Step 3.4.2.1.3.1.1

Multiply $7$ by $7$ .

$y = 49 + 7 x + x \cdot 7 + x \cdot x$

Step 3.4.2.1.3.1.2

Move $7$ to the left of $x$ .

$y = 49 + 7 x + 7 \cdot x + x \cdot x$

Step 3.4.2.1.3.1.3

Multiply $x$ by $x$ .

$y = 49 + 7 x + 7 x + x^{2}$

Step 3.4.2.1.3.2

Add $7 x$ and $7 x$ .

$y = 49 + 14 x + x^{2}$

Step 3.5

Simplify $49 + 14 x + x^{2}$ .

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Step 3.5.1

Move $49$ .

$y = 14 x + x^{2} + 49$

Step 3.5.2

Reorder $14 x$ and $x^{2}$ .

$y = x^{2} + 14 x + 49$

Step 4

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

$f^{- 1} (x) = x^{2} + 14 x + 49$

Step 5

Verify if $f^{- 1} (x) = x^{2} + 14 x + 49$ is the inverse of $f (x) = x^{\frac{1}{2}} - 7$ .

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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 $f^{- 1} (x^{\frac{1}{2}} - 7)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (x^{\frac{1}{2}} - 7) = {(x^{\frac{1}{2}} - 7)}^{2} + 14 (x^{\frac{1}{2}} - 7) + 49$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Rewrite ${(x^{\frac{1}{2}} - 7)}^{2}$ as $(x^{\frac{1}{2}} - 7) (x^{\frac{1}{2}} - 7)$ .

$f^{- 1} (x^{\frac{1}{2}} - 7) = (x^{\frac{1}{2}} - 7) (x^{\frac{1}{2}} - 7) + 14 (x^{\frac{1}{2}} - 7) + 49$

Step 5.2.3.2

Expand $(x^{\frac{1}{2}} - 7) (x^{\frac{1}{2}} - 7)$ using the FOIL Method.

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Step 5.2.3.2.1

Apply the distributive property.

$f^{- 1} (x^{\frac{1}{2}} - 7) = x^{\frac{1}{2}} (x^{\frac{1}{2}} - 7) - 7 (x^{\frac{1}{2}} - 7) + 14 (x^{\frac{1}{2}} - 7) + 49$

Step 5.2.3.2.2

Apply the distributive property.

$f^{- 1} (x^{\frac{1}{2}} - 7) = x^{\frac{1}{2}} x^{\frac{1}{2}} + x^{\frac{1}{2}} \cdot - 7 - 7 (x^{\frac{1}{2}} - 7) + 14 (x^{\frac{1}{2}} - 7) + 49$

Step 5.2.3.2.3

Apply the distributive property.

$f^{- 1} (x^{\frac{1}{2}} - 7) = x^{\frac{1}{2}} x^{\frac{1}{2}} + x^{\frac{1}{2}} \cdot - 7 - 7 x^{\frac{1}{2}} - 7 \cdot - 7 + 14 (x^{\frac{1}{2}} - 7) + 49$

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 $x^{\frac{1}{2}}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 5.2.3.3.1.1.1

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

$f^{- 1} (x^{\frac{1}{2}} - 7) = x^{\frac{1}{2} + \frac{1}{2}} + x^{\frac{1}{2}} \cdot - 7 - 7 x^{\frac{1}{2}} - 7 \cdot - 7 + 14 (x^{\frac{1}{2}} - 7) + 49$

Step 5.2.3.3.1.1.2

Combine the numerators over the common denominator.

$f^{- 1} (x^{\frac{1}{2}} - 7) = x^{\frac{1 + 1}{2}} + x^{\frac{1}{2}} \cdot - 7 - 7 x^{\frac{1}{2}} - 7 \cdot - 7 + 14 (x^{\frac{1}{2}} - 7) + 49$

Step 5.2.3.3.1.1.3

Add $1$ and $1$ .

$f^{- 1} (x^{\frac{1}{2}} - 7) = x^{\frac{2}{2}} + x^{\frac{1}{2}} \cdot - 7 - 7 x^{\frac{1}{2}} - 7 \cdot - 7 + 14 (x^{\frac{1}{2}} - 7) + 49$

Step 5.2.3.3.1.1.4

Divide $2$ by $2$ .

$f^{- 1} (x^{\frac{1}{2}} - 7) = x + x^{\frac{1}{2}} \cdot - 7 - 7 x^{\frac{1}{2}} - 7 \cdot - 7 + 14 (x^{\frac{1}{2}} - 7) + 49$

Step 5.2.3.3.1.2

Simplify $x^{1}$ .

$f^{- 1} (x^{\frac{1}{2}} - 7) = x + x^{\frac{1}{2}} \cdot - 7 - 7 x^{\frac{1}{2}} - 7 \cdot - 7 + 14 (x^{\frac{1}{2}} - 7) + 49$

Step 5.2.3.3.1.3

Move $- 7$ to the left of $x^{\frac{1}{2}}$ .

$f^{- 1} (x^{\frac{1}{2}} - 7) = x - 7 \cdot x^{\frac{1}{2}} - 7 x^{\frac{1}{2}} - 7 \cdot - 7 + 14 (x^{\frac{1}{2}} - 7) + 49$

Step 5.2.3.3.1.4

Multiply $- 7$ by $- 7$ .

$f^{- 1} (x^{\frac{1}{2}} - 7) = x - 7 x^{\frac{1}{2}} - 7 x^{\frac{1}{2}} + 49 + 14 (x^{\frac{1}{2}} - 7) + 49$

Step 5.2.3.3.2

Subtract $7 x^{\frac{1}{2}}$ from $- 7 x^{\frac{1}{2}}$ .

$f^{- 1} (x^{\frac{1}{2}} - 7) = x - 14 x^{\frac{1}{2}} + 49 + 14 (x^{\frac{1}{2}} - 7) + 49$

Step 5.2.3.4

Apply the distributive property.

$f^{- 1} (x^{\frac{1}{2}} - 7) = x - 14 x^{\frac{1}{2}} + 49 + 14 x^{\frac{1}{2}} + 14 \cdot - 7 + 49$

Step 5.2.3.5

Multiply $14$ by $- 7$ .

$f^{- 1} (x^{\frac{1}{2}} - 7) = x - 14 x^{\frac{1}{2}} + 49 + 14 x^{\frac{1}{2}} - 98 + 49$

Step 5.2.4

Simplify by adding terms.

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Step 5.2.4.1

Combine the opposite terms in $x - 14 x^{\frac{1}{2}} + 49 + 14 x^{\frac{1}{2}} - 98 + 49$ .

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Step 5.2.4.1.1

Add $- 14 x^{\frac{1}{2}}$ and $14 x^{\frac{1}{2}}$ .

$f^{- 1} (x^{\frac{1}{2}} - 7) = x + 0 + 49 - 98 + 49$

Step 5.2.4.1.2

Add $x$ and $0$ .

$f^{- 1} (x^{\frac{1}{2}} - 7) = x + 49 - 98 + 49$

Step 5.2.4.2

Subtract $98$ from $49$ .

$f^{- 1} (x^{\frac{1}{2}} - 7) = x - 49 + 49$

Step 5.2.4.3

Combine the opposite terms in $x - 49 + 49$ .

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Step 5.2.4.3.1

Add $- 49$ and $49$ .

$f^{- 1} (x^{\frac{1}{2}} - 7) = x + 0$

Step 5.2.4.3.2

Add $x$ and $0$ .

$f^{- 1} (x^{\frac{1}{2}} - 7) = x$

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 $f (x^{2} + 14 x + 49)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x^{2} + 14 x + 49) = {(x^{2} + 14 x + 49)}^{\frac{1}{2}} - 7$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Factor using the perfect square rule.

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Step 5.3.3.1.1

Rewrite $49$ as $7^{2}$ .

$f (x^{2} + 14 x + 49) = {(x^{2} + 14 x + 7^{2})}^{\frac{1}{2}} - 7$

Step 5.3.3.1.2

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

$14 x = 2 \cdot x \cdot 7$

Step 5.3.3.1.3

Rewrite the polynomial.

$f (x^{2} + 14 x + 49) = {(x^{2} + 2 \cdot x \cdot 7 + 7^{2})}^{\frac{1}{2}} - 7$

Step 5.3.3.1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 7$ .

$f (x^{2} + 14 x + 49) = {({(x + 7)}^{2})}^{\frac{1}{2}} - 7$

Step 5.3.3.2

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

$f (x^{2} + 14 x + 49) = {(x + 7)}^{2 (\frac{1}{2})} - 7$

Step 5.3.3.3

Cancel the common factor of $2$ .

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Step 5.3.3.3.1

Cancel the common factor.

$f (x^{2} + 14 x + 49) = {(x + 7)}^{2 (\frac{1}{2})} - 7$

Step 5.3.3.3.2

Rewrite the expression.

$f (x^{2} + 14 x + 49) = (x + 7) - 7$

Step 5.3.3.4

Simplify.

$f (x^{2} + 14 x + 49) = x + 7 - 7$

Step 5.3.4

Combine the opposite terms in $x + 7 - 7$ .

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Step 5.3.4.1

Subtract $7$ from $7$ .

$f (x^{2} + 14 x + 49) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (x^{2} + 14 x + 49) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x^{2} + 14 x + 49$ is the inverse of $f (x) = x^{\frac{1}{2}} - 7$ .

$f^{- 1} (x) = x^{2} + 14 x + 49$