Precalculus Examples

$f (x) = - 4 + 6 \sqrt{7 x + 4}$

Step 1

Write $f (x) = - 4 + 6 \sqrt{7 x + 4}$ as an equation.

$y = - 4 + 6 \sqrt{7 x + 4}$

Step 2

Interchange the variables.

$x = - 4 + 6 \sqrt{7 y + 4}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $- 4 + 6 \sqrt{7 y + 4} = x$ .

$- 4 + 6 \sqrt{7 y + 4} = x$

Step 3.2

Add $4$ to both sides of the equation.

$6 \sqrt{7 y + 4} = x + 4$

Step 3.3

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

${(6 \sqrt{7 y + 4})}^{2} = {(x + 4)}^{2}$

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 $\sqrt{7 y + 4}$ as ${(7 y + 4)}^{\frac{1}{2}}$ .

${(6 {(7 y + 4)}^{\frac{1}{2}})}^{2} = {(x + 4)}^{2}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Simplify ${(6 {(7 y + 4)}^{\frac{1}{2}})}^{2}$ .

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Step 3.4.2.1.1

Apply the product rule to $6 {(7 y + 4)}^{\frac{1}{2}}$ .

$6^{2} {({(7 y + 4)}^{\frac{1}{2}})}^{2} = {(x + 4)}^{2}$

Step 3.4.2.1.2

Raise $6$ to the power of $2$ .

$36 {({(7 y + 4)}^{\frac{1}{2}})}^{2} = {(x + 4)}^{2}$

Step 3.4.2.1.3

Multiply the exponents in ${({(7 y + 4)}^{\frac{1}{2}})}^{2}$ .

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Step 3.4.2.1.3.1

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

$36 {(7 y + 4)}^{\frac{1}{2} \cdot 2} = {(x + 4)}^{2}$

Step 3.4.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.4.2.1.3.2.1

Cancel the common factor.

$36 {(7 y + 4)}^{\frac{1}{2} \cdot 2} = {(x + 4)}^{2}$

Step 3.4.2.1.3.2.2

Rewrite the expression.

$36 {(7 y + 4)}^{1} = {(x + 4)}^{2}$

Step 3.4.2.1.4

Simplify.

$36 (7 y + 4) = {(x + 4)}^{2}$

Step 3.4.2.1.5

Apply the distributive property.

$36 (7 y) + 36 \cdot 4 = {(x + 4)}^{2}$

Step 3.4.2.1.6

Multiply.

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Step 3.4.2.1.6.1

Multiply $7$ by $36$ .

$252 y + 36 \cdot 4 = {(x + 4)}^{2}$

Step 3.4.2.1.6.2

Multiply $36$ by $4$ .

$252 y + 144 = {(x + 4)}^{2}$

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 $(x + 4) (x + 4)$ .

$252 y + 144 = (x + 4) (x + 4)$

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.

$252 y + 144 = x (x + 4) + 4 (x + 4)$

Step 3.4.3.1.2.2

Apply the distributive property.

$252 y + 144 = x \cdot x + x \cdot 4 + 4 (x + 4)$

Step 3.4.3.1.2.3

Apply the distributive property.

$252 y + 144 = x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4$

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 $x$ .

$252 y + 144 = x^{2} + x \cdot 4 + 4 x + 4 \cdot 4$

Step 3.4.3.1.3.1.2

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

$252 y + 144 = x^{2} + 4 \cdot x + 4 x + 4 \cdot 4$

Step 3.4.3.1.3.1.3

Multiply $4$ by $4$ .

$252 y + 144 = x^{2} + 4 x + 4 x + 16$

Step 3.4.3.1.3.2

Add $4 x$ and $4 x$ .

$252 y + 144 = x^{2} + 8 x + 16$

Step 3.5

Solve for $y$ .

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Step 3.5.1

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

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Step 3.5.1.1

Subtract $144$ from both sides of the equation.

$252 y = x^{2} + 8 x + 16 - 144$

Step 3.5.1.2

Subtract $144$ from $16$ .

$252 y = x^{2} + 8 x - 128$

Step 3.5.2

Divide each term in $252 y = x^{2} + 8 x - 128$ by $252$ and simplify.

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Step 3.5.2.1

Divide each term in $252 y = x^{2} + 8 x - 128$ by $252$ .

$\frac{252 y}{252} = \frac{x^{2}}{252} + \frac{8 x}{252} + \frac{- 128}{252}$

Step 3.5.2.2

Simplify the left side.

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Step 3.5.2.2.1

Cancel the common factor of $252$ .

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Step 3.5.2.2.1.1

Cancel the common factor.

$\frac{252 y}{252} = \frac{x^{2}}{252} + \frac{8 x}{252} + \frac{- 128}{252}$

Step 3.5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{2}}{252} + \frac{8 x}{252} + \frac{- 128}{252}$

Step 3.5.2.3

Simplify the right side.

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Step 3.5.2.3.1

Simplify each term.

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Step 3.5.2.3.1.1

Cancel the common factor of $8$ and $252$ .

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Step 3.5.2.3.1.1.1

Factor $4$ out of $8 x$ .

$y = \frac{x^{2}}{252} + \frac{4 (2 x)}{252} + \frac{- 128}{252}$

Step 3.5.2.3.1.1.2

Cancel the common factors.

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Step 3.5.2.3.1.1.2.1

Factor $4$ out of $252$ .

$y = \frac{x^{2}}{252} + \frac{4 (2 x)}{4 (63)} + \frac{- 128}{252}$

Step 3.5.2.3.1.1.2.2

Cancel the common factor.

$y = \frac{x^{2}}{252} + \frac{4 (2 x)}{4 \cdot 63} + \frac{- 128}{252}$

Step 3.5.2.3.1.1.2.3

Rewrite the expression.

$y = \frac{x^{2}}{252} + \frac{2 x}{63} + \frac{- 128}{252}$

Step 3.5.2.3.1.2

Cancel the common factor of $- 128$ and $252$ .

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Step 3.5.2.3.1.2.1

Factor $4$ out of $- 128$ .

$y = \frac{x^{2}}{252} + \frac{2 x}{63} + \frac{4 (- 32)}{252}$

Step 3.5.2.3.1.2.2

Cancel the common factors.

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Step 3.5.2.3.1.2.2.1

Factor $4$ out of $252$ .

$y = \frac{x^{2}}{252} + \frac{2 x}{63} + \frac{4 \cdot - 32}{4 \cdot 63}$

Step 3.5.2.3.1.2.2.2

Cancel the common factor.

$y = \frac{x^{2}}{252} + \frac{2 x}{63} + \frac{4 \cdot - 32}{4 \cdot 63}$

Step 3.5.2.3.1.2.2.3

Rewrite the expression.

$y = \frac{x^{2}}{252} + \frac{2 x}{63} + \frac{- 32}{63}$

Step 3.5.2.3.1.3

Move the negative in front of the fraction.

$y = \frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}$

Step 4

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

$f^{- 1} (x) = \frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}$

Step 5

Verify if $f^{- 1} (x) = \frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}$ is the inverse of $f (x) = - 4 + 6 \sqrt{7 x + 4}$ .

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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 $f^{- 1} (- 4 + 6 \sqrt{7 x + 4})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{{(- 4 + 6 \sqrt{7 x + 4})}^{2}}{252} + \frac{2 (- 4 + 6 \sqrt{7 x + 4})}{63} - \frac{32}{63}$

Step 5.2.3

Combine the numerators over the common denominator.

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{{(- 4 + 6 \sqrt{7 x + 4})}^{2}}{252} + \frac{2 (- 4 + 6 \sqrt{7 x + 4}) - 32}{63}$

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

Apply the distributive property.

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{{(- 4 + 6 \sqrt{7 x + 4})}^{2}}{252} + \frac{2 \cdot - 4 + 2 (6 \sqrt{7 x + 4}) - 32}{63}$

Step 5.2.4.2

Multiply $2$ by $- 4$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{{(- 4 + 6 \sqrt{7 x + 4})}^{2}}{252} + \frac{- 8 + 2 (6 \sqrt{7 x + 4}) - 32}{63}$

Step 5.2.4.3

Multiply $6$ by $2$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{{(- 4 + 6 \sqrt{7 x + 4})}^{2}}{252} + \frac{- 8 + 12 \sqrt{7 x + 4} - 32}{63}$

Step 5.2.5

Subtract $32$ from $- 8$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{{(- 4 + 6 \sqrt{7 x + 4})}^{2}}{252} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6

Simplify each term.

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Step 5.2.6.1

Simplify the numerator.

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Step 5.2.6.1.1

Factor $2$ out of $- 4 + 6 \sqrt{7 x + 4}$ .

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Step 5.2.6.1.1.1

Factor $2$ out of $- 4$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{{(2 (- 2) + 6 \sqrt{7 x + 4})}^{2}}{252} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.1.1.2

Factor $2$ out of $6 \sqrt{7 x + 4}$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{{(2 (- 2) + 2 (3 \sqrt{7 x + 4}))}^{2}}{252} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.1.1.3

Factor $2$ out of $2 (- 2) + 2 (3 \sqrt{7 x + 4})$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{{(2 (- 2 + 3 \sqrt{7 x + 4}))}^{2}}{252} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.1.2

Apply the product rule to $2 (- 2 + 3 \sqrt{7 x + 4})$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{2^{2} {(- 2 + 3 \sqrt{7 x + 4})}^{2}}{252} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.1.3

Raise $2$ to the power of $2$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{4 {(- 2 + 3 \sqrt{7 x + 4})}^{2}}{252} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.2

Cancel the common factor of $4$ and $252$ .

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Step 5.2.6.2.1

Factor $4$ out of $4 {(- 2 + 3 \sqrt{7 x + 4})}^{2}$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{4 ({(- 2 + 3 \sqrt{7 x + 4})}^{2})}{252} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.2.2

Cancel the common factors.

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Step 5.2.6.2.2.1

Factor $4$ out of $252$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{4 {(- 2 + 3 \sqrt{7 x + 4})}^{2}}{4 \cdot 63} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.2.2.2

Cancel the common factor.

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{4 {(- 2 + 3 \sqrt{7 x + 4})}^{2}}{4 \cdot 63} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.2.2.3

Rewrite the expression.

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{{(- 2 + 3 \sqrt{7 x + 4})}^{2}}{63} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.3

Rewrite ${(- 2 + 3 \sqrt{7 x + 4})}^{2}$ as $(- 2 + 3 \sqrt{7 x + 4}) (- 2 + 3 \sqrt{7 x + 4})$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{(- 2 + 3 \sqrt{7 x + 4}) (- 2 + 3 \sqrt{7 x + 4})}{63} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.4

Expand $(- 2 + 3 \sqrt{7 x + 4}) (- 2 + 3 \sqrt{7 x + 4})$ using the FOIL Method.

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Step 5.2.6.4.1

Apply the distributive property.

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{- 2 (- 2 + 3 \sqrt{7 x + 4}) + 3 \sqrt{7 x + 4} (- 2 + 3 \sqrt{7 x + 4})}{63} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.4.2

Apply the distributive property.

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{- 2 \cdot - 2 - 2 (3 \sqrt{7 x + 4}) + 3 \sqrt{7 x + 4} (- 2 + 3 \sqrt{7 x + 4})}{63} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.4.3

Apply the distributive property.

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{- 2 \cdot - 2 - 2 (3 \sqrt{7 x + 4}) + 3 \sqrt{7 x + 4} \cdot - 2 + 3 \sqrt{7 x + 4} (3 \sqrt{7 x + 4})}{63} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.5

Simplify and combine like terms.

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Step 5.2.6.5.1

Simplify each term.

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Step 5.2.6.5.1.1

Multiply $- 2$ by $- 2$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{4 - 2 (3 \sqrt{7 x + 4}) + 3 \sqrt{7 x + 4} \cdot - 2 + 3 \sqrt{7 x + 4} (3 \sqrt{7 x + 4})}{63} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.5.1.2

Multiply $3$ by $- 2$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{4 - 6 \sqrt{7 x + 4} + 3 \sqrt{7 x + 4} \cdot - 2 + 3 \sqrt{7 x + 4} (3 \sqrt{7 x + 4})}{63} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.5.1.3

Multiply $- 2$ by $3$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{4 - 6 \sqrt{7 x + 4} - 6 \sqrt{7 x + 4} + 3 \sqrt{7 x + 4} (3 \sqrt{7 x + 4})}{63} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.5.1.4

Multiply $3 \sqrt{7 x + 4} (3 \sqrt{7 x + 4})$ .

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Step 5.2.6.5.1.4.1

Multiply $3$ by $3$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{4 - 6 \sqrt{7 x + 4} - 6 \sqrt{7 x + 4} + 9 \sqrt{7 x + 4} \sqrt{7 x + 4}}{63} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.5.1.4.2

Raise $\sqrt{7 x + 4}$ to the power of $1$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{4 - 6 \sqrt{7 x + 4} - 6 \sqrt{7 x + 4} + 9 (\sqrt{7 x + 4} \sqrt{7 x + 4})}{63} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.5.1.4.3

Raise $\sqrt{7 x + 4}$ to the power of $1$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{4 - 6 \sqrt{7 x + 4} - 6 \sqrt{7 x + 4} + 9 (\sqrt{7 x + 4} \sqrt{7 x + 4})}{63} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.5.1.4.4

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

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{4 - 6 \sqrt{7 x + 4} - 6 \sqrt{7 x + 4} + 9 {\sqrt{7 x + 4}}^{1 + 1}}{63} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.5.1.4.5

Add $1$ and $1$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{4 - 6 \sqrt{7 x + 4} - 6 \sqrt{7 x + 4} + 9 {\sqrt{7 x + 4}}^{2}}{63} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.5.1.5

Rewrite ${\sqrt{7 x + 4}}^{2}$ as $7 x + 4$ .

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Step 5.2.6.5.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 x + 4}$ as ${(7 x + 4)}^{\frac{1}{2}}$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{4 - 6 \sqrt{7 x + 4} - 6 \sqrt{7 x + 4} + 9 {({(7 x + 4)}^{\frac{1}{2}})}^{2}}{63} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.5.1.5.2

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

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{4 - 6 \sqrt{7 x + 4} - 6 \sqrt{7 x + 4} + 9 {(7 x + 4)}^{\frac{1}{2} \cdot 2}}{63} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.5.1.5.3

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

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{4 - 6 \sqrt{7 x + 4} - 6 \sqrt{7 x + 4} + 9 {(7 x + 4)}^{\frac{2}{2}}}{63} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.5.1.5.4

Cancel the common factor of $2$ .

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Step 5.2.6.5.1.5.4.1

Cancel the common factor.

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{4 - 6 \sqrt{7 x + 4} - 6 \sqrt{7 x + 4} + 9 {(7 x + 4)}^{\frac{2}{2}}}{63} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.5.1.5.4.2

Rewrite the expression.

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{4 - 6 \sqrt{7 x + 4} - 6 \sqrt{7 x + 4} + 9 (7 x + 4)}{63} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.5.1.5.5

Simplify.

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{4 - 6 \sqrt{7 x + 4} - 6 \sqrt{7 x + 4} + 9 (7 x + 4)}{63} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.5.1.6

Apply the distributive property.

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{4 - 6 \sqrt{7 x + 4} - 6 \sqrt{7 x + 4} + 9 (7 x) + 9 \cdot 4}{63} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.5.1.7

Multiply $7$ by $9$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{4 - 6 \sqrt{7 x + 4} - 6 \sqrt{7 x + 4} + 63 x + 9 \cdot 4}{63} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.5.1.8

Multiply $9$ by $4$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{4 - 6 \sqrt{7 x + 4} - 6 \sqrt{7 x + 4} + 63 x + 36}{63} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.5.2

Add $4$ and $36$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{40 - 6 \sqrt{7 x + 4} - 6 \sqrt{7 x + 4} + 63 x}{63} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.5.3

Subtract $6 \sqrt{7 x + 4}$ from $- 6 \sqrt{7 x + 4}$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{40 - 12 \sqrt{7 x + 4} + 63 x}{63} + \frac{- 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.6

Factor $4$ out of $- 40 + 12 \sqrt{7 x + 4}$ .

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Step 5.2.6.6.1

Factor $4$ out of $- 40$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{40 - 12 \sqrt{7 x + 4} + 63 x}{63} + \frac{4 (- 10) + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.6.6.2

Factor $4$ out of $12 \sqrt{7 x + 4}$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{40 - 12 \sqrt{7 x + 4} + 63 x}{63} + \frac{4 (- 10) + 4 (3 \sqrt{7 x + 4})}{63}$

Step 5.2.6.6.3

Factor $4$ out of $4 (- 10) + 4 (3 \sqrt{7 x + 4})$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{40 - 12 \sqrt{7 x + 4} + 63 x}{63} + \frac{4 (- 10 + 3 \sqrt{7 x + 4})}{63}$

Step 5.2.7

Combine the numerators over the common denominator.

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{40 - 12 \sqrt{7 x + 4} + 63 x + 4 (- 10 + 3 \sqrt{7 x + 4})}{63}$

Step 5.2.8

Simplify each term.

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Step 5.2.8.1

Apply the distributive property.

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{40 - 12 \sqrt{7 x + 4} + 63 x + 4 \cdot - 10 + 4 (3 \sqrt{7 x + 4})}{63}$

Step 5.2.8.2

Multiply $4$ by $- 10$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{40 - 12 \sqrt{7 x + 4} + 63 x - 40 + 4 (3 \sqrt{7 x + 4})}{63}$

Step 5.2.8.3

Multiply $3$ by $4$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{40 - 12 \sqrt{7 x + 4} + 63 x - 40 + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.9

Simplify terms.

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Step 5.2.9.1

Combine the opposite terms in $40 - 12 \sqrt{7 x + 4} + 63 x - 40 + 12 \sqrt{7 x + 4}$ .

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Step 5.2.9.1.1

Subtract $40$ from $40$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{0 - 12 \sqrt{7 x + 4} + 63 x + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.9.1.2

Subtract $12 \sqrt{7 x + 4}$ from $0$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{- 12 \sqrt{7 x + 4} + 63 x + 12 \sqrt{7 x + 4}}{63}$

Step 5.2.9.1.3

Add $- 12 \sqrt{7 x + 4}$ and $12 \sqrt{7 x + 4}$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{63 x + 0}{63}$

Step 5.2.9.1.4

Add $63 x$ and $0$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{63 x}{63}$

Step 5.2.9.2

Cancel the common factor of $63$ .

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Step 5.2.9.2.1

Cancel the common factor.

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = \frac{63 x}{63}$

Step 5.2.9.2.2

Divide $x$ by $1$ .

$f^{- 1} (- 4 + 6 \sqrt{7 x + 4}) = x$

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 $f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{7 (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) + 4}$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{7 (\frac{x^{2}}{252}) + 7 (\frac{2 x}{63}) + 7 (- \frac{32}{63}) + 4}$

Step 5.3.3.2

Simplify.

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Step 5.3.3.2.1

Cancel the common factor of $7$ .

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Step 5.3.3.2.1.1

Factor $7$ out of $252$ .

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{7 (\frac{x^{2}}{7 (36)}) + 7 (\frac{2 x}{63}) + 7 (- \frac{32}{63}) + 4}$

Step 5.3.3.2.1.2

Cancel the common factor.

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{7 (\frac{x^{2}}{7 \cdot 36}) + 7 (\frac{2 x}{63}) + 7 (- \frac{32}{63}) + 4}$

Step 5.3.3.2.1.3

Rewrite the expression.

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{\frac{x^{2}}{36} + 7 (\frac{2 x}{63}) + 7 (- \frac{32}{63}) + 4}$

Step 5.3.3.2.2

Cancel the common factor of $7$ .

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Step 5.3.3.2.2.1

Factor $7$ out of $63$ .

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{\frac{x^{2}}{36} + 7 (\frac{2 x}{7 (9)}) + 7 (- \frac{32}{63}) + 4}$

Step 5.3.3.2.2.2

Cancel the common factor.

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{\frac{x^{2}}{36} + 7 (\frac{2 x}{7 \cdot 9}) + 7 (- \frac{32}{63}) + 4}$

Step 5.3.3.2.2.3

Rewrite the expression.

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{\frac{x^{2}}{36} + \frac{2 x}{9} + 7 (- \frac{32}{63}) + 4}$

Step 5.3.3.2.3

Cancel the common factor of $7$ .

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Step 5.3.3.2.3.1

Move the leading negative in $- \frac{32}{63}$ into the numerator.

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{\frac{x^{2}}{36} + \frac{2 x}{9} + 7 (\frac{- 32}{63}) + 4}$

Step 5.3.3.2.3.2

Factor $7$ out of $63$ .

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{\frac{x^{2}}{36} + \frac{2 x}{9} + 7 (\frac{- 32}{7 (9)}) + 4}$

Step 5.3.3.2.3.3

Cancel the common factor.

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{\frac{x^{2}}{36} + \frac{2 x}{9} + 7 (\frac{- 32}{7 \cdot 9}) + 4}$

Step 5.3.3.2.3.4

Rewrite the expression.

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{\frac{x^{2}}{36} + \frac{2 x}{9} + \frac{- 32}{9} + 4}$

Step 5.3.3.3

Move the negative in front of the fraction.

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{\frac{x^{2}}{36} + \frac{2 x}{9} - \frac{32}{9} + 4}$

Step 5.3.3.4

To write $4$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{\frac{x^{2}}{36} + \frac{2 x}{9} - \frac{32}{9} + 4 \cdot \frac{9}{9}}$

Step 5.3.3.5

Combine $4$ and $\frac{9}{9}$ .

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{\frac{x^{2}}{36} + \frac{2 x}{9} - \frac{32}{9} + \frac{4 \cdot 9}{9}}$

Step 5.3.3.6

Combine the numerators over the common denominator.

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{\frac{x^{2}}{36} + \frac{2 x}{9} + \frac{- 32 + 4 \cdot 9}{9}}$

Step 5.3.3.7

Simplify the numerator.

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Step 5.3.3.7.1

Multiply $4$ by $9$ .

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{\frac{x^{2}}{36} + \frac{2 x}{9} + \frac{- 32 + 36}{9}}$

Step 5.3.3.7.2

Add $- 32$ and $36$ .

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{\frac{x^{2}}{36} + \frac{2 x}{9} + \frac{4}{9}}$

Step 5.3.3.8

Factor using the perfect square rule.

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Step 5.3.3.8.1

Rewrite $36$ as $6^{2}$ .

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{\frac{x^{2}}{6^{2}} + \frac{2 x}{9} + \frac{4}{9}}$

Step 5.3.3.8.2

Rewrite $\frac{x^{2}}{6^{2}}$ as ${(\frac{x}{6})}^{2}$ .

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{{(\frac{x}{6})}^{2} + \frac{2 x}{9} + \frac{4}{9}}$

Step 5.3.3.8.3

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

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{{(\frac{x}{6})}^{2} + \frac{2 x}{9} + \frac{2^{2}}{9}}$

Step 5.3.3.8.4

Rewrite $9$ as $3^{2}$ .

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{{(\frac{x}{6})}^{2} + \frac{2 x}{9} + \frac{2^{2}}{3^{2}}}$

Step 5.3.3.8.5

Rewrite $\frac{2^{2}}{3^{2}}$ as ${(\frac{2}{3})}^{2}$ .

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{{(\frac{x}{6})}^{2} + \frac{2 x}{9} + {(\frac{2}{3})}^{2}}$

Step 5.3.3.8.6

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

$\frac{2 x}{9} = 2 \cdot \frac{x}{6} \cdot \frac{2}{3}$

Step 5.3.3.8.7

Rewrite the polynomial.

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{{(\frac{x}{6})}^{2} + 2 \cdot \frac{x}{6} \cdot \frac{2}{3} + {(\frac{2}{3})}^{2}}$

Step 5.3.3.8.8

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = \frac{x}{6}$ and $b = \frac{2}{3}$ .

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{{(\frac{x}{6} + \frac{2}{3})}^{2}}$

Step 5.3.3.9

To write $\frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{{(\frac{x}{6} + \frac{2}{3} \cdot \frac{2}{2})}^{2}}$

Step 5.3.3.10

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.3.10.1

Multiply $\frac{2}{3}$ by $\frac{2}{2}$ .

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{{(\frac{x}{6} + \frac{2 \cdot 2}{3 \cdot 2})}^{2}}$

Step 5.3.3.10.2

Multiply $3$ by $2$ .

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{{(\frac{x}{6} + \frac{2 \cdot 2}{6})}^{2}}$

Step 5.3.3.11

Combine the numerators over the common denominator.

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{{(\frac{x + 2 \cdot 2}{6})}^{2}}$

Step 5.3.3.12

Multiply $2$ by $2$ .

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{{(\frac{x + 4}{6})}^{2}}$

Step 5.3.3.13

Apply the product rule to $\frac{x + 4}{6}$ .

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{\frac{{(x + 4)}^{2}}{6^{2}}}$

Step 5.3.3.14

Raise $6$ to the power of $2$ .

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{\frac{{(x + 4)}^{2}}{36}}$

Step 5.3.3.15

Rewrite $36$ as $6^{2}$ .

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{\frac{{(x + 4)}^{2}}{6^{2}}}$

Step 5.3.3.16

Rewrite $\frac{{(x + 4)}^{2}}{6^{2}}$ as ${(\frac{x + 4}{6})}^{2}$ .

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 \sqrt{{(\frac{x + 4}{6})}^{2}}$

Step 5.3.3.17

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

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 (\frac{x + 4}{6})$

Step 5.3.3.18

Cancel the common factor of $6$ .

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Step 5.3.3.18.1

Cancel the common factor.

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + 6 (\frac{x + 4}{6})$

Step 5.3.3.18.2

Rewrite the expression.

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = - 4 + x + 4$

Step 5.3.4

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

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Step 5.3.4.1

Add $- 4$ and $4$ .

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (\frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}$ is the inverse of $f (x) = - 4 + 6 \sqrt{7 x + 4}$ .

$f^{- 1} (x) = \frac{x^{2}}{252} + \frac{2 x}{63} - \frac{32}{63}$