Find the Inverse 5y+4=(x+3)^2+1/2

$5 y + 4 = {(x + 3)}^{2} + \frac{1}{2}$

Simplify ${(x + 3)}^{2} + \frac{1}{2}$ .

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Simplify each term.

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Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$5 y + 4 = (x + 3) (x + 3) + \frac{1}{2}$

Expand $(x + 3) (x + 3)$ using the FOIL Method.

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Apply the distributive property.

$5 y + 4 = x (x + 3) + 3 (x + 3) + \frac{1}{2}$

Apply the distributive property.

$5 y + 4 = x \cdot x + x \cdot 3 + 3 (x + 3) + \frac{1}{2}$

Apply the distributive property.

$5 y + 4 = x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3 + \frac{1}{2}$

Simplify and combine like terms.

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Simplify each term.

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Multiply $x$ by $x$ .

$5 y + 4 = x^{2} + x \cdot 3 + 3 x + 3 \cdot 3 + \frac{1}{2}$

Move $3$ to the left of $x$ .

$5 y + 4 = x^{2} + 3 \cdot x + 3 x + 3 \cdot 3 + \frac{1}{2}$

Multiply $3$ by $3$ .

$5 y + 4 = x^{2} + 3 x + 3 x + 9 + \frac{1}{2}$

Add $3 x$ and $3 x$ .

$5 y + 4 = x^{2} + 6 x + 9 + \frac{1}{2}$

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

$5 y + 4 = x^{2} + 6 x + 9 \cdot \frac{2}{2} + \frac{1}{2}$

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

$5 y + 4 = x^{2} + 6 x + \frac{9 \cdot 2}{2} + \frac{1}{2}$

Combine the numerators over the common denominator.

$5 y + 4 = x^{2} + 6 x + \frac{9 \cdot 2 + 1}{2}$

Simplify the numerator.

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Multiply $9$ by $2$ .

$5 y + 4 = x^{2} + 6 x + \frac{18 + 1}{2}$

Add $18$ and $1$ .

$5 y + 4 = x^{2} + 6 x + \frac{19}{2}$

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

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Subtract $4$ from both sides of the equation.

$5 y = x^{2} + 6 x + \frac{19}{2} - 4$

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

$5 y = x^{2} + 6 x + \frac{19}{2} - 4 \cdot \frac{2}{2}$

Combine $- 4$ and $\frac{2}{2}$ .

$5 y = x^{2} + 6 x + \frac{19}{2} + \frac{- 4 \cdot 2}{2}$

Combine the numerators over the common denominator.

$5 y = x^{2} + 6 x + \frac{19 - 4 \cdot 2}{2}$

Simplify the numerator.

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Multiply $- 4$ by $2$ .

$5 y = x^{2} + 6 x + \frac{19 - 8}{2}$

Subtract $8$ from $19$ .

$5 y = x^{2} + 6 x + \frac{11}{2}$

Divide each term by $5$ and simplify.

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Divide each term in $5 y = x^{2} + 6 x + \frac{11}{2}$ by $5$ .

$\frac{5 y}{5} = \frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{2} \cdot \frac{1}{5}$

Cancel the common factor of $5$ .

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Cancel the common factor.

$\frac{5 y}{5} = \frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{2} \cdot \frac{1}{5}$

Divide $y$ by $1$ .

$y = \frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{2} \cdot \frac{1}{5}$

Multiply $\frac{11}{2} \cdot \frac{1}{5}$ .

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Multiply $\frac{11}{2}$ and $\frac{1}{5}$ .

$y = \frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{2 \cdot 5}$

Multiply $2$ by $5$ .

$y = \frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}$

Interchange the variables.

$x = \frac{y^{2}}{5} + \frac{6 y}{5} + \frac{11}{10}$

Solve for $y$ .

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Rewrite the equation as $\frac{y^{2}}{5} + \frac{6 y}{5} + \frac{11}{10} = x$ .

$\frac{y^{2}}{5} + \frac{6 y}{5} + \frac{11}{10} = x$

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

$\frac{y^{2}}{5} + \frac{6 y}{5} + \frac{11}{10} - x = 0$

Multiply through by the least common denominator $10$ , then simplify.

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Apply the distributive property.

$10 (\frac{y^{2}}{5}) + 10 (\frac{6 y}{5}) + 10 (\frac{11}{10}) + 10 (- x) = 0$

Simplify.

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Cancel the common factor of $5$ .

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Factor $5$ out of $10$ .

$5 (2) (\frac{y^{2}}{5}) + 10 (\frac{6 y}{5}) + 10 (\frac{11}{10}) + 10 (- x) = 0$

Cancel the common factor.

$5 \cdot (2 (\frac{y^{2}}{5})) + 10 (\frac{6 y}{5}) + 10 (\frac{11}{10}) + 10 (- x) = 0$

Rewrite the expression.

$2 y^{2} + 10 (\frac{6 y}{5}) + 10 (\frac{11}{10}) + 10 (- x) = 0$

Cancel the common factor of $5$ .

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Factor $5$ out of $10$ .

$2 y^{2} + 5 (2) (\frac{6 y}{5}) + 10 (\frac{11}{10}) + 10 (- x) = 0$

Cancel the common factor.

$2 y^{2} + 5 \cdot (2 (\frac{6 y}{5})) + 10 (\frac{11}{10}) + 10 (- x) = 0$

Rewrite the expression.

$2 y^{2} + 2 (6 y) + 10 (\frac{11}{10}) + 10 (- x) = 0$

Multiply $6$ by $2$ .

$2 y^{2} + 12 y + 10 (\frac{11}{10}) + 10 (- x) = 0$

Cancel the common factor of $10$ .

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Cancel the common factor.

$2 y^{2} + 12 y + 10 (\frac{11}{10}) + 10 (- x) = 0$

Rewrite the expression.

$2 y^{2} + 12 y + 11 + 10 (- x) = 0$

Multiply $- 1$ by $10$ .

$2 y^{2} + 12 y + 11 - 10 x = 0$

Move $11$ .

$2 y^{2} + 12 y - 10 x + 11 = 0$

Move $12 y$ .

$2 y^{2} - 10 x + 12 y + 11 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 2$ , $b = 12$ , and $c = - 10 x + 11$ into the quadratic formula and solve for $y$ .

$\frac{- 12 \pm \sqrt{12^{2} - 4 \cdot (2 \cdot (- 10 x + 11))}}{2 \cdot 2}$

Simplify.

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Simplify the numerator.

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Raise $12$ to the power of $2$ .

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot (2 \cdot (- 10 x + 11))}}{2 \cdot 2}$

Apply the distributive property.

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot (2 (- 10 x) + 2 \cdot 11)}}{2 \cdot 2}$

Multiply $- 10$ by $2$ .

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot (- 20 x + 2 \cdot 11)}}{2 \cdot 2}$

Multiply $2$ by $11$ .

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot (- 20 x + 22)}}{2 \cdot 2}$

Apply the distributive property.

$y = \frac{- 12 \pm \sqrt{144 - 4 (- 20 x) - 4 \cdot 22}}{2 \cdot 2}$

Multiply $- 20$ by $- 4$ .

$y = \frac{- 12 \pm \sqrt{144 + 80 x - 4 \cdot 22}}{2 \cdot 2}$

Multiply $- 4$ by $22$ .

$y = \frac{- 12 \pm \sqrt{144 + 80 x - 88}}{2 \cdot 2}$

Subtract $88$ from $144$ .

$y = \frac{- 12 \pm \sqrt{80 x + 56}}{2 \cdot 2}$

Factor $8$ out of $80 x + 56$ .

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Factor $8$ out of $80 x$ .

$y = \frac{- 12 \pm \sqrt{8 (10 x) + 56}}{2 \cdot 2}$

Factor $8$ out of $56$ .

$y = \frac{- 12 \pm \sqrt{8 (10 x) + 8 (7)}}{2 \cdot 2}$

Factor $8$ out of $8 (10 x) + 8 (7)$ .

$y = \frac{- 12 \pm \sqrt{8 (10 x + 7)}}{2 \cdot 2}$

Rewrite $8 (10 x + 7)$ as $2^{2} \cdot (2 (10 x + 7))$ .

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Factor $4$ out of $8$ .

$y = \frac{- 12 \pm \sqrt{4 (2) (10 x + 7)}}{2 \cdot 2}$

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

$y = \frac{- 12 \pm \sqrt{2^{2} \cdot (2 (10 x + 7))}}{2 \cdot 2}$

Add parentheses.

$y = \frac{- 12 \pm \sqrt{2^{2} \cdot (2 (10 x + 7))}}{2 \cdot 2}$

Pull terms out from under the radical.

$y = \frac{- 12 \pm 2 \sqrt{2 (10 x + 7)}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$y = \frac{- 12 \pm 2 \sqrt{2 (10 x + 7)}}{4}$

Simplify $\frac{- 12 \pm 2 \sqrt{2 (10 x + 7)}}{4}$ .

$y = \frac{- 6 \pm \sqrt{2 (10 x + 7)}}{2}$

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

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Simplify the numerator.

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Raise $12$ to the power of $2$ .

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot (2 \cdot (- 10 x + 11))}}{2 \cdot 2}$

Apply the distributive property.

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot (2 (- 10 x) + 2 \cdot 11)}}{2 \cdot 2}$

Multiply $- 10$ by $2$ .

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot (- 20 x + 2 \cdot 11)}}{2 \cdot 2}$

Multiply $2$ by $11$ .

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot (- 20 x + 22)}}{2 \cdot 2}$

Apply the distributive property.

$y = \frac{- 12 \pm \sqrt{144 - 4 (- 20 x) - 4 \cdot 22}}{2 \cdot 2}$

Multiply $- 20$ by $- 4$ .

$y = \frac{- 12 \pm \sqrt{144 + 80 x - 4 \cdot 22}}{2 \cdot 2}$

Multiply $- 4$ by $22$ .

$y = \frac{- 12 \pm \sqrt{144 + 80 x - 88}}{2 \cdot 2}$

Subtract $88$ from $144$ .

$y = \frac{- 12 \pm \sqrt{80 x + 56}}{2 \cdot 2}$

Factor $8$ out of $80 x + 56$ .

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Factor $8$ out of $80 x$ .

$y = \frac{- 12 \pm \sqrt{8 (10 x) + 56}}{2 \cdot 2}$

Factor $8$ out of $56$ .

$y = \frac{- 12 \pm \sqrt{8 (10 x) + 8 (7)}}{2 \cdot 2}$

Factor $8$ out of $8 (10 x) + 8 (7)$ .

$y = \frac{- 12 \pm \sqrt{8 (10 x + 7)}}{2 \cdot 2}$

Rewrite $8 (10 x + 7)$ as $2^{2} \cdot (2 (10 x + 7))$ .

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Factor $4$ out of $8$ .

$y = \frac{- 12 \pm \sqrt{4 (2) (10 x + 7)}}{2 \cdot 2}$

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

$y = \frac{- 12 \pm \sqrt{2^{2} \cdot (2 (10 x + 7))}}{2 \cdot 2}$

Add parentheses.

$y = \frac{- 12 \pm \sqrt{2^{2} \cdot (2 (10 x + 7))}}{2 \cdot 2}$

Pull terms out from under the radical.

$y = \frac{- 12 \pm 2 \sqrt{2 (10 x + 7)}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$y = \frac{- 12 \pm 2 \sqrt{2 (10 x + 7)}}{4}$

Simplify $\frac{- 12 \pm 2 \sqrt{2 (10 x + 7)}}{4}$ .

$y = \frac{- 6 \pm \sqrt{2 (10 x + 7)}}{2}$

Change the $\pm$ to $+$ .

$y = \frac{- 6 + \sqrt{2 (10 x + 7)}}{2}$

Rewrite $- 6$ as $- 1 (6)$ .

$y = \frac{- 1 \cdot 6 + \sqrt{2 (10 x + 7)}}{2}$

Factor $- 1$ out of $\sqrt{2 (10 x + 7)}$ .

$y = \frac{- 1 \cdot 6 - 1 (- \sqrt{2 (10 x + 7)})}{2}$

Factor $- 1$ out of $- 1 (6) - 1 (- \sqrt{2 (10 x + 7)})$ .

$y = \frac{- 1 (6 - \sqrt{2 (10 x + 7)})}{2}$

Move the negative in front of the fraction.

$y = - \frac{6 - \sqrt{2 (10 x + 7)}}{2}$

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

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Simplify the numerator.

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Raise $12$ to the power of $2$ .

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot (2 \cdot (- 10 x + 11))}}{2 \cdot 2}$

Apply the distributive property.

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot (2 (- 10 x) + 2 \cdot 11)}}{2 \cdot 2}$

Multiply $- 10$ by $2$ .

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot (- 20 x + 2 \cdot 11)}}{2 \cdot 2}$

Multiply $2$ by $11$ .

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot (- 20 x + 22)}}{2 \cdot 2}$

Apply the distributive property.

$y = \frac{- 12 \pm \sqrt{144 - 4 (- 20 x) - 4 \cdot 22}}{2 \cdot 2}$

Multiply $- 20$ by $- 4$ .

$y = \frac{- 12 \pm \sqrt{144 + 80 x - 4 \cdot 22}}{2 \cdot 2}$

Multiply $- 4$ by $22$ .

$y = \frac{- 12 \pm \sqrt{144 + 80 x - 88}}{2 \cdot 2}$

Subtract $88$ from $144$ .

$y = \frac{- 12 \pm \sqrt{80 x + 56}}{2 \cdot 2}$

Factor $8$ out of $80 x + 56$ .

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Factor $8$ out of $80 x$ .

$y = \frac{- 12 \pm \sqrt{8 (10 x) + 56}}{2 \cdot 2}$

Factor $8$ out of $56$ .

$y = \frac{- 12 \pm \sqrt{8 (10 x) + 8 (7)}}{2 \cdot 2}$

Factor $8$ out of $8 (10 x) + 8 (7)$ .

$y = \frac{- 12 \pm \sqrt{8 (10 x + 7)}}{2 \cdot 2}$

Rewrite $8 (10 x + 7)$ as $2^{2} \cdot (2 (10 x + 7))$ .

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Factor $4$ out of $8$ .

$y = \frac{- 12 \pm \sqrt{4 (2) (10 x + 7)}}{2 \cdot 2}$

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

$y = \frac{- 12 \pm \sqrt{2^{2} \cdot (2 (10 x + 7))}}{2 \cdot 2}$

Add parentheses.

$y = \frac{- 12 \pm \sqrt{2^{2} \cdot (2 (10 x + 7))}}{2 \cdot 2}$

Pull terms out from under the radical.

$y = \frac{- 12 \pm 2 \sqrt{2 (10 x + 7)}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$y = \frac{- 12 \pm 2 \sqrt{2 (10 x + 7)}}{4}$

Simplify $\frac{- 12 \pm 2 \sqrt{2 (10 x + 7)}}{4}$ .

$y = \frac{- 6 \pm \sqrt{2 (10 x + 7)}}{2}$

Change the $\pm$ to $-$ .

$y = \frac{- 6 - \sqrt{2 (10 x + 7)}}{2}$

Factor $- 1$ out of $- 6 - \sqrt{2 (10 x + 7)}$ .

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Rewrite $- 6$ as $- 1 (6)$ .

$y = \frac{- 1 \cdot 6 - \sqrt{2 (10 x + 7)}}{2}$

Factor $- 1$ out of $- \sqrt{2 (10 x + 7)}$ .

$y = \frac{- 1 \cdot 6 - (\sqrt{2 (10 x + 7)})}{2}$

Factor $- 1$ out of $- 1 (6) - (\sqrt{2 (10 x + 7)})$ .

$y = \frac{- 1 (6 + \sqrt{2 (10 x + 7)})}{2}$

Rewrite $- 1 (6 + \sqrt{2 (10 x + 7)})$ as $- (6 + \sqrt{2 (10 x + 7)})$ .

$y = \frac{- (6 + \sqrt{2 (10 x + 7)})}{2}$

Move the negative in front of the fraction.

$y = - \frac{6 + \sqrt{2 (10 x + 7)}}{2}$

The final answer is the combination of both solutions.

$y = - \frac{6 - \sqrt{2 (10 x + 7)}}{2}$

$y = - \frac{6 + \sqrt{2 (10 x + 7)}}{2}$

$y = - \frac{6 - \sqrt{2 (10 x + 7)}}{2}$

$y = - \frac{6 + \sqrt{2 (10 x + 7)}}{2}$

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.

$f^{- 1} (x) = - \frac{6 - \sqrt{2 (10 x + 7)}}{2}, - \frac{6 + \sqrt{2 (10 x + 7)}}{2}$

Set up the composite result function.

$g (f (x))$

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

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - \sqrt{2 (10 (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) + 7)}}{2}$

Simplify the numerator.

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Apply the distributive property.

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - \sqrt{2 (10 (\frac{x^{2}}{5}) + 10 (\frac{6 x}{5}) + 10 (\frac{11}{10}) + 7)}}{2}$

Simplify.

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Cancel the common factor of $5$ .

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Factor $5$ out of $10$ .

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - \sqrt{2 (5 (2) (\frac{x^{2}}{5}) + 10 (\frac{6 x}{5}) + 10 (\frac{11}{10}) + 7)}}{2}$

Cancel the common factor.

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - \sqrt{2 (5 \cdot (2 (\frac{x^{2}}{5})) + 10 (\frac{6 x}{5}) + 10 (\frac{11}{10}) + 7)}}{2}$

Rewrite the expression.

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - \sqrt{2 (2 x^{2} + 10 (\frac{6 x}{5}) + 10 (\frac{11}{10}) + 7)}}{2}$

Cancel the common factor of $5$ .

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Factor $5$ out of $10$ .

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - \sqrt{2 (2 x^{2} + 5 (2) (\frac{6 x}{5}) + 10 (\frac{11}{10}) + 7)}}{2}$

Cancel the common factor.

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - \sqrt{2 (2 x^{2} + 5 \cdot (2 (\frac{6 x}{5})) + 10 (\frac{11}{10}) + 7)}}{2}$

Rewrite the expression.

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - \sqrt{2 (2 x^{2} + 2 (6 x) + 10 (\frac{11}{10}) + 7)}}{2}$

Multiply $6$ by $2$ .

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - \sqrt{2 (2 x^{2} + 12 x + 10 (\frac{11}{10}) + 7)}}{2}$

Cancel the common factor of $10$ .

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Cancel the common factor.

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - \sqrt{2 (2 x^{2} + 12 x + 10 (\frac{11}{10}) + 7)}}{2}$

Rewrite the expression.

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - \sqrt{2 (2 x^{2} + 12 x + 11 + 7)}}{2}$

Add $11$ and $7$ .

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - \sqrt{2 (2 x^{2} + 12 x + 18)}}{2}$

Rewrite $2 x^{2} + 12 x + 18$ in a factored form.

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Factor $2$ out of $2 x^{2} + 12 x + 18$ .

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Factor $2$ out of $2 x^{2}$ .

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - \sqrt{2 (2 (x^{2}) + 12 x + 18)}}{2}$

Factor $2$ out of $12 x$ .

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - \sqrt{2 (2 (x^{2}) + 2 (6 x) + 18)}}{2}$

Factor $2$ out of $18$ .

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - \sqrt{2 (2 x^{2} + 2 (6 x) + 2 \cdot 9)}}{2}$

Factor $2$ out of $2 x^{2} + 2 (6 x)$ .

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - \sqrt{2 (2 (x^{2} + 6 x) + 2 \cdot 9)}}{2}$

Factor $2$ out of $2 (x^{2} + 6 x) + 2 \cdot 9$ .

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - \sqrt{2 (2 (x^{2} + 6 x + 9))}}{2}$

Factor using the perfect square rule.

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

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - \sqrt{2 (2 (x^{2} + 6 x + 3^{2}))}}{2}$

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

$2 a b = 2 \cdot x \cdot 3$

Simplify.

$2 a b = 6 x$

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

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - \sqrt{2 (2 {(x + 3)}^{2})}}{2}$

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - \sqrt{2 \cdot (2 {(x + 3)}^{2})}}{2}$

Multiply $2$ by $2$ .

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - \sqrt{4 {(x + 3)}^{2}}}{2}$

Rewrite $4 {(x + 3)}^{2}$ as ${(2 (x + 3))}^{2}$ .

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - \sqrt{{(2 (x + 3))}^{2}}}{2}$

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

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - (2 (x + 3))}{2}$

Apply the distributive property.

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - (2 x + 2 \cdot 3)}{2}$

Multiply $2$ by $3$ .

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - (2 x + 6)}{2}$

Apply the distributive property.

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - (2 x) - 1 \cdot 6}{2}$

Multiply $2$ by $- 1$ .

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - 2 x - 1 \cdot 6}{2}$

Multiply $- 1$ by $6$ .

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{6 - 2 x - 6}{2}$

Subtract $6$ from $6$ .

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{- 2 x + 0}{2}$

Add $- 2 x$ and $0$ .

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{- 2 x}{2}$

Cancel the common factor of $- 2$ and $2$ .

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

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{2 (- x)}{2}$

Cancel the common factors.

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Factor $2$ out of $2$ .

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{2 (- x)}{2 (1)}$

Cancel the common factor.

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{2 (- x)}{2 \cdot 1}$

Rewrite the expression.

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = - \frac{- x}{1}$

Divide $- x$ by $1$ .

$g (\frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}) = x$

Since $g (f (x)) = x$ , $f^{- 1} (x) = - \frac{6 - \sqrt{2 (10 x + 7)}}{2}, - \frac{6 + \sqrt{2 (10 x + 7)}}{2}$ is the inverse of $f (x) = \frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}$ .

$f^{- 1} (x) = - \frac{6 - \sqrt{2 (10 x + 7)}}{2}, - \frac{6 + \sqrt{2 (10 x + 7)}}{2}$

Find the Inverse 5y+4=(x+3)^2+1/2

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JackHudson

Charlie Trom

Lucy Evel