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

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

Step 1

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

Simplify each term.

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.

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.

Simplify each term.

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.

Multiply $9$ by $2$ .

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

Add $18$ and $1$ .

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

Step 2

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

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.

Multiply $- 4$ by $2$ .

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

Subtract $8$ from $19$ .

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

Step 3

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

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

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

Multiply $2$ by $5$ .

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

Step 4

Interchange the variables.

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

Step 5

Solve for $y$ .

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

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

Subtract $x$ from both sides of the equation.

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

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

Apply the distributive property.

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

Simplify.

Cancel the common factor of $5$ .

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 $5$ .

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 $10$ .

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.

Simplify the numerator.

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

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

Multiply $- 4$ by $2$ .

$y = \frac{- 12 \pm \sqrt{144 - 8 \cdot (- 10 x + 11)}}{2 \cdot 2}$

Apply the distributive property.

$y = \frac{- 12 \pm \sqrt{144 - 8 (- 10 x) - 8 \cdot 11}}{2 \cdot 2}$

Multiply $- 10$ by $- 8$ .

$y = \frac{- 12 \pm \sqrt{144 + 80 x - 8 \cdot 11}}{2 \cdot 2}$

Multiply $- 8$ by $11$ .

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

Subtract $88$ from $144$ .

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

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

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

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 $\pm$ .

Simplify the numerator.

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

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

Multiply $- 4$ by $2$ .

$y = \frac{- 12 \pm \sqrt{144 - 8 \cdot (- 10 x + 11)}}{2 \cdot 2}$

Apply the distributive property.

$y = \frac{- 12 \pm \sqrt{144 - 8 (- 10 x) - 8 \cdot 11}}{2 \cdot 2}$

Multiply $- 10$ by $- 8$ .

$y = \frac{- 12 \pm \sqrt{144 + 80 x - 8 \cdot 11}}{2 \cdot 2}$

Multiply $- 8$ by $11$ .

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

Subtract $88$ from $144$ .

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

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

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

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 $\pm$ .

Simplify the numerator.

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

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

Multiply $- 4$ by $2$ .

$y = \frac{- 12 \pm \sqrt{144 - 8 \cdot (- 10 x + 11)}}{2 \cdot 2}$

Apply the distributive property.

$y = \frac{- 12 \pm \sqrt{144 - 8 (- 10 x) - 8 \cdot 11}}{2 \cdot 2}$

Multiply $- 10$ by $- 8$ .

$y = \frac{- 12 \pm \sqrt{144 + 80 x - 8 \cdot 11}}{2 \cdot 2}$

Multiply $- 8$ by $11$ .

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

Subtract $88$ from $144$ .

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

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

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

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

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.

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

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

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

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

Step 6

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

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

Step 7

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

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = \frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}$ and $f^{- 1} (x) = - \frac{6 - \sqrt{2 (10 x + 7)}}{2}, - \frac{6 + \sqrt{2 (10 x + 7)}}{2}$ and compare them.

Find the range of $f (x) = \frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}$ .

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[- \frac{7}{10}, \infty)$

Find the domain of $- \frac{6 - \sqrt{2 (10 x + 7)}}{2}$ .

Set the radicand in $\sqrt{2 (10 x + 7)}$ greater than or equal to $0$ to find where the expression is defined.

$2 (10 x + 7) \geq 0$

Solve for $x$ .

Divide each term in $2 (10 x + 7) \geq 0$ by $2$ and simplify.

Divide each term in $2 (10 x + 7) \geq 0$ by $2$ .

$\frac{2 (10 x + 7)}{2} \geq \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 (10 x + 7)}{2} \geq \frac{0}{2}$

Divide $10 x + 7$ by $1$ .

$10 x + 7 \geq \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$10 x + 7 \geq 0$

Subtract $7$ from both sides of the inequality.

$10 x \geq - 7$

Divide each term in $10 x \geq - 7$ by $10$ and simplify.

Divide each term in $10 x \geq - 7$ by $10$ .

$\frac{10 x}{10} \geq \frac{- 7}{10}$

Simplify the left side.

Cancel the common factor of $10$ .

Cancel the common factor.

$\frac{10 x}{10} \geq \frac{- 7}{10}$

Divide $x$ by $1$ .

$x \geq \frac{- 7}{10}$

Simplify the right side.

Move the negative in front of the fraction.

$x \geq - \frac{7}{10}$

The domain is all values of $x$ that make the expression defined.

$[- \frac{7}{10}, \infty)$

Find the domain of $f (x) = \frac{x^{2}}{5} + \frac{6 x}{5} + \frac{11}{10}$ .

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

$(- \infty, \infty)$

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

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

Step 8

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

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