Find the Inverse (x^2)/8+(3x^2)/2+(9x)/2

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Trigonometry

$\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}$

Interchange the variables.

$x = \frac{y^{2}}{8} + \frac{3 y^{2}}{2} + \frac{9 y}{2}$

Solve for $y$ .

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Rewrite the equation as $\frac{y^{2}}{8} + \frac{3 y^{2}}{2} + \frac{9 y}{2} = x$ .

$\frac{y^{2}}{8} + \frac{3 y^{2}}{2} + \frac{9 y}{2} = x$

Simplify $\frac{y^{2}}{8} + \frac{3 y^{2}}{2} + \frac{9 y}{2}$ .

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To write $\frac{3 y^{2}}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{y^{2}}{8} + \frac{3 y^{2}}{2} \cdot \frac{4}{4} + \frac{9 y}{2} = x$

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{3 y^{2}}{2}$ and $\frac{4}{4}$ .

$\frac{y^{2}}{8} + \frac{3 y^{2} \cdot 4}{2 \cdot 4} + \frac{9 y}{2} = x$

Multiply $2$ by $4$ .

$\frac{y^{2}}{8} + \frac{3 y^{2} \cdot 4}{8} + \frac{9 y}{2} = x$

Combine the numerators over the common denominator.

$\frac{y^{2} + 3 y^{2} \cdot 4}{8} + \frac{9 y}{2} = x$

Simplify each term.

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Simplify the numerator.

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Factor $y^{2}$ out of $y^{2} + 3 y^{2} \cdot 4$ .

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Multiply by $1$ .

$\frac{y^{2} \cdot 1 + 3 y^{2} \cdot 4}{8} + \frac{9 y}{2} = x$

Factor $y^{2}$ out of $3 y^{2} \cdot 4$ .

$\frac{y^{2} \cdot 1 + y^{2} (3 \cdot 4)}{8} + \frac{9 y}{2} = x$

Factor $y^{2}$ out of $y^{2} \cdot 1 + y^{2} (3 \cdot 4)$ .

$\frac{y^{2} (1 + 3 \cdot 4)}{8} + \frac{9 y}{2} = x$

Multiply $3$ by $4$ .

$\frac{y^{2} (1 + 12)}{8} + \frac{9 y}{2} = x$

Add $1$ and $12$ .

$\frac{y^{2} \cdot 13}{8} + \frac{9 y}{2} = x$

Move $13$ to the left of $y^{2}$ .

$\frac{13 y^{2}}{8} + \frac{9 y}{2} = x$

Move $x$ to the left side of the equation by subtracting it from both sides.

$\frac{13 y^{2}}{8} + \frac{9 y}{2} - x = 0$

Multiply through by the least common denominator $8$ , then simplify.

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Apply the distributive property.

$8 (\frac{13 y^{2}}{8}) + 8 (\frac{9 y}{2}) + 8 (- x) = 0$

Simplify.

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Cancel the common factor of $8$ .

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Cancel the common factor.

$8 (\frac{13 y^{2}}{8}) + 8 (\frac{9 y}{2}) + 8 (- x) = 0$

Rewrite the expression.

$13 y^{2} + 8 (\frac{9 y}{2}) + 8 (- x) = 0$

Cancel the common factor of $2$ .

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Factor $2$ out of $8$ .

$13 y^{2} + 2 (4) (\frac{9 y}{2}) + 8 (- x) = 0$

Cancel the common factor.

$13 y^{2} + 2 \cdot (4 (\frac{9 y}{2})) + 8 (- x) = 0$

Rewrite the expression.

$13 y^{2} + 4 (9 y) + 8 (- x) = 0$

Multiply $9$ by $4$ .

$13 y^{2} + 36 y + 8 (- x) = 0$

Multiply $- 1$ by $8$ .

$13 y^{2} + 36 y - 8 x = 0$

Move $36 y$ .

$13 y^{2} - 8 x + 36 y = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 13$ , $b = 36$ , and $c = - 8 x$ into the quadratic formula and solve for $y$ .

$\frac{- 36 \pm \sqrt{36^{2} - 4 \cdot (13 \cdot (- 8 x))}}{2 \cdot 13}$

Simplify.

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Simplify the numerator.

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Raise $36$ to the power of $2$ .

$y = \frac{- 36 \pm \sqrt{1296 - 4 \cdot (13 \cdot (- 8 x))}}{2 \cdot 13}$

Multiply $- 8$ by $13$ .

$y = \frac{- 36 \pm \sqrt{1296 - 4 \cdot (- 104 x)}}{2 \cdot 13}$

Multiply $- 104$ by $- 4$ .

$y = \frac{- 36 \pm \sqrt{1296 + 416 x}}{2 \cdot 13}$

Factor $16$ out of $1296 + 416 x$ .

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Factor $16$ out of $1296$ .

$y = \frac{- 36 \pm \sqrt{16 (81) + 416 x}}{2 \cdot 13}$

Factor $16$ out of $416 x$ .

$y = \frac{- 36 \pm \sqrt{16 (81) + 16 (26 x)}}{2 \cdot 13}$

Factor $16$ out of $16 (81) + 16 (26 x)$ .

$y = \frac{- 36 \pm \sqrt{16 (81 + 26 x)}}{2 \cdot 13}$

Rewrite $16 (81 + 26 x)$ as ${(2^{2})}^{2} (81 + 26 x)$ .

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

$y = \frac{- 36 \pm \sqrt{4^{2} (81 + 26 x)}}{2 \cdot 13}$

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

$y = \frac{- 36 \pm \sqrt{{(2^{2})}^{2} (81 + 26 x)}}{2 \cdot 13}$

Pull terms out from under the radical.

$y = \frac{- 36 \pm 2^{2} \sqrt{81 + 26 x}}{2 \cdot 13}$

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

$y = \frac{- 36 \pm 4 \sqrt{81 + 26 x}}{2 \cdot 13}$

Multiply $2$ by $13$ .

$y = \frac{- 36 \pm 4 \sqrt{81 + 26 x}}{26}$

Simplify $\frac{- 36 \pm 4 \sqrt{81 + 26 x}}{26}$ .

$y = \frac{- 18 \pm 2 \sqrt{81 + 26 x}}{13}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Simplify the numerator.

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Raise $36$ to the power of $2$ .

$y = \frac{- 36 \pm \sqrt{1296 - 4 \cdot (13 \cdot (- 8 x))}}{2 \cdot 13}$

Multiply $- 8$ by $13$ .

$y = \frac{- 36 \pm \sqrt{1296 - 4 \cdot (- 104 x)}}{2 \cdot 13}$

Multiply $- 104$ by $- 4$ .

$y = \frac{- 36 \pm \sqrt{1296 + 416 x}}{2 \cdot 13}$

Factor $16$ out of $1296 + 416 x$ .

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Factor $16$ out of $1296$ .

$y = \frac{- 36 \pm \sqrt{16 (81) + 416 x}}{2 \cdot 13}$

Factor $16$ out of $416 x$ .

$y = \frac{- 36 \pm \sqrt{16 (81) + 16 (26 x)}}{2 \cdot 13}$

Factor $16$ out of $16 (81) + 16 (26 x)$ .

$y = \frac{- 36 \pm \sqrt{16 (81 + 26 x)}}{2 \cdot 13}$

Rewrite $16 (81 + 26 x)$ as ${(2^{2})}^{2} (81 + 26 x)$ .

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

$y = \frac{- 36 \pm \sqrt{4^{2} (81 + 26 x)}}{2 \cdot 13}$

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

$y = \frac{- 36 \pm \sqrt{{(2^{2})}^{2} (81 + 26 x)}}{2 \cdot 13}$

Pull terms out from under the radical.

$y = \frac{- 36 \pm 2^{2} \sqrt{81 + 26 x}}{2 \cdot 13}$

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

$y = \frac{- 36 \pm 4 \sqrt{81 + 26 x}}{2 \cdot 13}$

Multiply $2$ by $13$ .

$y = \frac{- 36 \pm 4 \sqrt{81 + 26 x}}{26}$

Simplify $\frac{- 36 \pm 4 \sqrt{81 + 26 x}}{26}$ .

$y = \frac{- 18 \pm 2 \sqrt{81 + 26 x}}{13}$

Change the $\pm$ to $+$ .

$y = \frac{- 18 + 2 \sqrt{81 + 26 x}}{13}$

Factor $2$ out of $- 18 + 2 \sqrt{81 + 26 x}$ .

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Factor $2$ out of $- 18$ .

$y = \frac{2 \cdot - 9 + 2 \sqrt{81 + 26 x}}{13}$

Factor $2$ out of $2 \cdot - 9 + 2 \sqrt{81 + 26 x}$ .

$y = \frac{2 (- 9 + \sqrt{81 + 26 x})}{13}$

Rewrite $- 9$ as $- 1 (9)$ .

$y = \frac{2 (- 1 \cdot 9 + \sqrt{81 + 26 x})}{13}$

Factor $- 1$ out of $\sqrt{81 + 26 x}$ .

$y = \frac{2 (- 1 \cdot 9 - 1 (- \sqrt{81 + 26 x}))}{13}$

Factor $- 1$ out of $- 1 (9) - 1 (- \sqrt{81 + 26 x})$ .

$y = \frac{2 (- 1 (9 - \sqrt{81 + 26 x}))}{13}$

Move the negative in front of the fraction.

$y = - \frac{2 (9 - \sqrt{81 + 26 x})}{13}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Simplify the numerator.

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Raise $36$ to the power of $2$ .

$y = \frac{- 36 \pm \sqrt{1296 - 4 \cdot (13 \cdot (- 8 x))}}{2 \cdot 13}$

Multiply $- 8$ by $13$ .

$y = \frac{- 36 \pm \sqrt{1296 - 4 \cdot (- 104 x)}}{2 \cdot 13}$

Multiply $- 104$ by $- 4$ .

$y = \frac{- 36 \pm \sqrt{1296 + 416 x}}{2 \cdot 13}$

Factor $16$ out of $1296 + 416 x$ .

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Factor $16$ out of $1296$ .

$y = \frac{- 36 \pm \sqrt{16 (81) + 416 x}}{2 \cdot 13}$

Factor $16$ out of $416 x$ .

$y = \frac{- 36 \pm \sqrt{16 (81) + 16 (26 x)}}{2 \cdot 13}$

Factor $16$ out of $16 (81) + 16 (26 x)$ .

$y = \frac{- 36 \pm \sqrt{16 (81 + 26 x)}}{2 \cdot 13}$

Rewrite $16 (81 + 26 x)$ as ${(2^{2})}^{2} (81 + 26 x)$ .

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

$y = \frac{- 36 \pm \sqrt{4^{2} (81 + 26 x)}}{2 \cdot 13}$

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

$y = \frac{- 36 \pm \sqrt{{(2^{2})}^{2} (81 + 26 x)}}{2 \cdot 13}$

Pull terms out from under the radical.

$y = \frac{- 36 \pm 2^{2} \sqrt{81 + 26 x}}{2 \cdot 13}$

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

$y = \frac{- 36 \pm 4 \sqrt{81 + 26 x}}{2 \cdot 13}$

Multiply $2$ by $13$ .

$y = \frac{- 36 \pm 4 \sqrt{81 + 26 x}}{26}$

Simplify $\frac{- 36 \pm 4 \sqrt{81 + 26 x}}{26}$ .

$y = \frac{- 18 \pm 2 \sqrt{81 + 26 x}}{13}$

Change the $\pm$ to $-$ .

$y = \frac{- 18 - 2 \sqrt{81 + 26 x}}{13}$

Factor $- 2$ out of $- 18 - 2 \sqrt{81 + 26 x}$ .

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Reorder $81$ and $26 x$ .

$y = \frac{- 18 - 2 \sqrt{26 x + 81}}{13}$

Factor $- 2$ out of $- 18$ .

$y = \frac{- 2 \cdot 9 - 2 \sqrt{26 x + 81}}{13}$

Factor $- 2$ out of $- 2 \sqrt{26 x + 81}$ .

$y = \frac{- 2 \cdot 9 - 2 \sqrt{26 x + 81}}{13}$

Factor $- 2$ out of $- 2 (9) - 2 (\sqrt{26 x + 81})$ .

$y = \frac{- 2 (9 + \sqrt{26 x + 81})}{13}$

Move the negative in front of the fraction.

$y = - \frac{2 (9 + \sqrt{26 x + 81})}{13}$

The final answer is the combination of both solutions.

$y = - \frac{2 (9 - \sqrt{81 + 26 x})}{13}$

$y = - \frac{2 (9 + \sqrt{26 x + 81})}{13}$

$y = - \frac{2 (9 - \sqrt{81 + 26 x})}{13}$

$y = - \frac{2 (9 + \sqrt{26 x + 81})}{13}$

Solve for $y$ and replace with $f^{- 1} (x)$ .

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

$f^{- 1} (x) = - \frac{2 (9 - \sqrt{81 + 26 x})}{13}, - \frac{2 (9 + \sqrt{26 x + 81})}{13}$

Set up the composite result function.

$g (f (x))$

Evaluate $g (f (x))$ by substituting in the value of $f$ into $g$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{81 + 26 (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2})})}{13}$

Simplify the numerator.

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To write $\frac{3 x^{2}}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{81 + 26 (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} \cdot \frac{4}{4} + \frac{9 x}{2})})}{13}$

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{3 x^{2}}{2}$ and $\frac{4}{4}$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{81 + 26 (\frac{x^{2}}{8} + \frac{3 x^{2} \cdot 4}{2 \cdot 4} + \frac{9 x}{2})})}{13}$

Multiply $2$ by $4$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{81 + 26 (\frac{x^{2}}{8} + \frac{3 x^{2} \cdot 4}{8} + \frac{9 x}{2})})}{13}$

Combine the numerators over the common denominator.

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{81 + 26 (\frac{x^{2} + 3 x^{2} \cdot 4}{8} + \frac{9 x}{2})})}{13}$

Simplify each term.

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Simplify the numerator.

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Factor $x^{2}$ out of $x^{2} + 3 x^{2} \cdot 4$ .

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Multiply by $1$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{81 + 26 (\frac{x^{2} \cdot 1 + 3 x^{2} \cdot 4}{8} + \frac{9 x}{2})})}{13}$

Factor $x^{2}$ out of $3 x^{2} \cdot 4$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{81 + 26 (\frac{x^{2} \cdot 1 + x^{2} (3 \cdot 4)}{8} + \frac{9 x}{2})})}{13}$

Factor $x^{2}$ out of $x^{2} \cdot 1 + x^{2} (3 \cdot 4)$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{81 + 26 (\frac{x^{2} (1 + 3 \cdot 4)}{8} + \frac{9 x}{2})})}{13}$

Multiply $3$ by $4$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{81 + 26 (\frac{x^{2} (1 + 12)}{8} + \frac{9 x}{2})})}{13}$

Add $1$ and $12$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{81 + 26 (\frac{x^{2} \cdot 13}{8} + \frac{9 x}{2})})}{13}$

Move $13$ to the left of $x^{2}$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{81 + 26 (\frac{13 x^{2}}{8} + \frac{9 x}{2})})}{13}$

Apply the distributive property.

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{81 + 26 (\frac{13 x^{2}}{8}) + 26 (\frac{9 x}{2})})}{13}$

Cancel the common factor of $2$ .

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Factor $2$ out of $26$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{81 + 2 (13) (\frac{13 x^{2}}{8}) + 26 (\frac{9 x}{2})})}{13}$

Factor $2$ out of $8$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{81 + 2 \cdot (13 (\frac{13 x^{2}}{2 \cdot 4})) + 26 (\frac{9 x}{2})})}{13}$

Cancel the common factor.

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{81 + 2 \cdot (13 (\frac{13 x^{2}}{2 \cdot 4})) + 26 (\frac{9 x}{2})})}{13}$

Rewrite the expression.

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{81 + 13 (\frac{13 x^{2}}{4}) + 26 (\frac{9 x}{2})})}{13}$

Combine $13$ and $\frac{13 x^{2}}{4}$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{81 + \frac{13 (13 x^{2})}{4} + 26 (\frac{9 x}{2})})}{13}$

Multiply $13$ by $13$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{81 + \frac{169 x^{2}}{4} + 26 (\frac{9 x}{2})})}{13}$

Cancel the common factor of $2$ .

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Factor $2$ out of $26$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{81 + \frac{169 x^{2}}{4} + 2 (13) (\frac{9 x}{2})})}{13}$

Cancel the common factor.

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{81 + \frac{169 x^{2}}{4} + 2 \cdot (13 (\frac{9 x}{2}))})}{13}$

Rewrite the expression.

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{81 + \frac{169 x^{2}}{4} + 13 (9 x)})}{13}$

Multiply $9$ by $13$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{81 + \frac{169 x^{2}}{4} + 117 x})}{13}$

Reorder terms.

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{\frac{169 x^{2}}{4} + 117 x + 81})}{13}$

Factor using the perfect square rule.

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Rewrite $169 x^{2}$ as ${(13 x)}^{2}$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{\frac{{(13 x)}^{2}}{4} + 117 x + 81})}{13}$

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

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{\frac{{(13 x)}^{2}}{2^{2}} + 117 x + 81})}{13}$

Rewrite $\frac{{(13 x)}^{2}}{2^{2}}$ as ${(\frac{13 x}{2})}^{2}$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{{(\frac{13 x}{2})}^{2} + 117 x + 81})}{13}$

Rewrite $81$ as $9^{2}$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{{(\frac{13 x}{2})}^{2} + 117 x + 9^{2}})}{13}$

Check the middle term by multiplying $2 a b$ and compare this result with the middle term in the original expression.

$2 a b = 2 \cdot \frac{13 x}{2} \cdot 9$

Simplify.

$2 a b = 117 x$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = \frac{13 x}{2}$ and $b = 9$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{{(\frac{13 x}{2} + 9)}^{2}})}{13}$

To write $9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{{(\frac{13 x}{2} + 9 \cdot \frac{2}{2})}^{2}})}{13}$

Combine $9$ and $\frac{2}{2}$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{{(\frac{13 x}{2} + \frac{9 \cdot 2}{2})}^{2}})}{13}$

Combine the numerators over the common denominator.

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{{(\frac{13 x + 9 \cdot 2}{2})}^{2}})}{13}$

Multiply $9$ by $2$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{{(\frac{13 x + 18}{2})}^{2}})}{13}$

Apply the product rule to $\frac{13 x + 18}{2}$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{\frac{{(13 x + 18)}^{2}}{2^{2}}})}{13}$

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

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{\frac{{(13 x + 18)}^{2}}{4}})}{13}$

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

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{\frac{{(13 x + 18)}^{2}}{2^{2}}})}{13}$

Rewrite $\frac{{(13 x + 18)}^{2}}{2^{2}}$ as ${(\frac{13 x + 18}{2})}^{2}$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \sqrt{{(\frac{13 x + 18}{2})}^{2}})}{13}$

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

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 - \frac{13 x + 18}{2})}{13}$

To write $9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (9 \cdot \frac{2}{2} - \frac{13 x + 18}{2})}{13}$

Combine $9$ and $\frac{2}{2}$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (\frac{9 \cdot 2}{2} - \frac{13 x + 18}{2})}{13}$

Combine the numerators over the common denominator.

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (\frac{9 \cdot 2 - (13 x + 18)}{2})}{13}$

Rewrite $\frac{9 \cdot 2 - (13 x + 18)}{2}$ in a factored form.

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Multiply $9$ by $2$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (\frac{18 - (13 x + 18)}{2})}{13}$

Apply the distributive property.

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (\frac{18 - (13 x) - 1 \cdot 18}{2})}{13}$

Multiply $13$ by $- 1$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (\frac{18 - 13 x - 1 \cdot 18}{2})}{13}$

Multiply $- 1$ by $18$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (\frac{18 - 13 x - 18}{2})}{13}$

Subtract $18$ from $18$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (\frac{- 13 x + 0}{2})}{13}$

Add $- 13 x$ and $0$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (\frac{- 13 x}{2})}{13}$

Move the negative in front of the fraction.

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{2 (- \frac{13 x}{2})}{13}$

Simplify the numerator.

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Multiply $2$ by $- 1$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{- 2 \frac{13 x}{2}}{13}$

Combine $- 2$ and $\frac{13 x}{2}$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{\frac{- 2 (13 x)}{2}}{13}$

Multiply $- 2$ by $13$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{\frac{- 26 x}{2}}{13}$

Reduce the expression by cancelling the common factors.

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Reduce the expression $\frac{- 26 x}{2}$ by cancelling the common factors.

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Factor $2$ out of $- 26 x$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{\frac{2 (- 13 x)}{2}}{13}$

Factor $2$ out of $2$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{\frac{2 (- 13 x)}{2 (1)}}{13}$

Cancel the common factor.

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{\frac{2 (- 13 x)}{2 \cdot 1}}{13}$

Rewrite the expression.

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{\frac{- 13 x}{1}}{13}$

Divide $- 13 x$ by $1$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{- 13 x}{13}$

Cancel the common factor of $- 13$ and $13$ .

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Factor $13$ out of $- 13 x$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{13 (- x)}{13}$

Cancel the common factors.

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Factor $13$ out of $13$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{13 (- x)}{13 (1)}$

Cancel the common factor.

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{13 (- x)}{13 \cdot 1}$

Rewrite the expression.

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = - \frac{- x}{1}$

Divide $- x$ by $1$ .

$g (\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}) = x$

Since $g (f (x)) = x$ , $f^{- 1} (x) = - \frac{2 (9 - \sqrt{81 + 26 x})}{13}, - \frac{2 (9 + \sqrt{26 x + 81})}{13}$ is the inverse of $f (x) = \frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}$ .

$f^{- 1} (x) = - \frac{2 (9 - \sqrt{81 + 26 x})}{13}, - \frac{2 (9 + \sqrt{26 x + 81})}{13}$

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