Algebra Examples

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

Step 1

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

Tap for more steps...

Step 1.1

Simplify each term.

Tap for more steps...

Step 1.1.1

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

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

Step 1.1.2

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

Tap for more steps...

Step 1.1.2.1

Apply the distributive property.

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

Step 1.1.2.2

Apply the distributive property.

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

Step 1.1.2.3

Apply the distributive property.

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

Step 1.1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.1.3.1

Simplify each term.

Tap for more steps...

Step 1.1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.1.3.1.2

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

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

Step 1.1.3.1.3

Multiply $3$ by $3$ .

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

Step 1.1.3.2

Add $3 x$ and $3 x$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 1.5

Simplify the numerator.

Tap for more steps...

Step 1.5.1

Multiply $9$ by $2$ .

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

Step 1.5.2

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.

Tap for more steps...

Step 2.1

Subtract $4$ from both sides of the equation.

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

Tap for more steps...

Step 2.5.1

Multiply $- 4$ by $2$ .

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

Step 2.5.2

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.

Tap for more steps...

Step 3.1

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

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

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

Divide $y$ by $1$ .

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

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Simplify each term.

Tap for more steps...

Step 3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.3.1.2

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

Tap for more steps...

Step 3.3.1.2.1

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

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

Step 3.3.1.2.2

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 $y$ .

Tap for more steps...

Step 5.1

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

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

Step 5.2

Subtract $x$ from both sides of the equation.

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

Step 5.3

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

Tap for more steps...

Step 5.3.1

Apply the distributive property.

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

Step 5.3.2

Simplify.

Tap for more steps...

Step 5.3.2.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.3.2.1.1

Factor $5$ out of $10$ .

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

Step 5.3.2.1.2

Cancel the common factor.

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

Step 5.3.2.1.3

Rewrite the expression.

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

Step 5.3.2.2

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.3.2.2.1

Factor $5$ out of $10$ .

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

Step 5.3.2.2.2

Cancel the common factor.

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

Step 5.3.2.2.3

Rewrite the expression.

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

Step 5.3.2.3

Multiply $6$ by $2$ .

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

Step 5.3.2.4

Cancel the common factor of $10$ .

Tap for more steps...

Step 5.3.2.4.1

Cancel the common factor.

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

Step 5.3.2.4.2

Rewrite the expression.

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

Step 5.3.2.5

Multiply $- 1$ by $10$ .

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

Step 5.3.3

Move $11$ .

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

Step 5.3.4

Move $12 y$ .

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

Step 5.4

Use the quadratic formula to find the solutions.

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

Step 5.5

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

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

Step 5.6

Simplify.

Tap for more steps...

Step 5.6.1

Simplify the numerator.

Tap for more steps...

Step 5.6.1.1

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

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

Step 5.6.1.2

Multiply $- 4$ by $2$ .

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

Step 5.6.1.3

Apply the distributive property.

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

Step 5.6.1.4

Multiply $- 10$ by $- 8$ .

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

Step 5.6.1.5

Multiply $- 8$ by $11$ .

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

Step 5.6.1.6

Subtract $88$ from $144$ .

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

Step 5.6.1.7

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

Tap for more steps...

Step 5.6.1.7.1

Factor $8$ out of $80 x$ .

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

Step 5.6.1.7.2

Factor $8$ out of $56$ .

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

Step 5.6.1.7.3

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

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

Step 5.6.1.8

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

Tap for more steps...

Step 5.6.1.8.1

Factor $4$ out of $8$ .

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

Step 5.6.1.8.2

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

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

Step 5.6.1.8.3

Add parentheses.

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

Step 5.6.1.9

Pull terms out from under the radical.

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

Step 5.6.2

Multiply $2$ by $2$ .

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

Step 5.6.3

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

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

Step 5.7

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

Tap for more steps...

Step 5.7.1

Simplify the numerator.

Tap for more steps...

Step 5.7.1.1

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

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

Step 5.7.1.2

Multiply $- 4$ by $2$ .

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

Step 5.7.1.3

Apply the distributive property.

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

Step 5.7.1.4

Multiply $- 10$ by $- 8$ .

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

Step 5.7.1.5

Multiply $- 8$ by $11$ .

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

Step 5.7.1.6

Subtract $88$ from $144$ .

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

Step 5.7.1.7

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

Tap for more steps...

Step 5.7.1.7.1

Factor $8$ out of $80 x$ .

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

Step 5.7.1.7.2

Factor $8$ out of $56$ .

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

Step 5.7.1.7.3

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

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

Step 5.7.1.8

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

Tap for more steps...

Step 5.7.1.8.1

Factor $4$ out of $8$ .

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

Step 5.7.1.8.2

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

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

Step 5.7.1.8.3

Add parentheses.

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

Step 5.7.1.9

Pull terms out from under the radical.

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

Step 5.7.2

Multiply $2$ by $2$ .

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

Step 5.7.3

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

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

Step 5.7.4

Change the $\pm$ to $+$ .

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

Step 5.7.5

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

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

Step 5.7.6

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

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

Step 5.7.7

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

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

Step 5.7.8

Move the negative in front of the fraction.

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

Step 5.8

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

Tap for more steps...

Step 5.8.1

Simplify the numerator.

Tap for more steps...

Step 5.8.1.1

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

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

Step 5.8.1.2

Multiply $- 4$ by $2$ .

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

Step 5.8.1.3

Apply the distributive property.

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

Step 5.8.1.4

Multiply $- 10$ by $- 8$ .

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

Step 5.8.1.5

Multiply $- 8$ by $11$ .

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

Step 5.8.1.6

Subtract $88$ from $144$ .

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

Step 5.8.1.7

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

Tap for more steps...

Step 5.8.1.7.1

Factor $8$ out of $80 x$ .

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

Step 5.8.1.7.2

Factor $8$ out of $56$ .

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

Step 5.8.1.7.3

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

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

Step 5.8.1.8

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

Tap for more steps...

Step 5.8.1.8.1

Factor $4$ out of $8$ .

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

Step 5.8.1.8.2

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

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

Step 5.8.1.8.3

Add parentheses.

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

Step 5.8.1.9

Pull terms out from under the radical.

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

Step 5.8.2

Multiply $2$ by $2$ .

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

Step 5.8.3

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

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

Step 5.8.4

Change the $\pm$ to $-$ .

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

Step 5.8.5

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

Tap for more steps...

Step 5.8.5.1

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

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

Step 5.8.5.2

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

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

Step 5.8.5.3

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

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

Step 5.8.5.4

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

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

Step 5.8.6

Move the negative in front of the fraction.

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

Step 5.9

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

Tap for more steps...

Step 7.1

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.

Step 7.2

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

Tap for more steps...

Step 7.2.1

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

Interval Notation:

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

Step 7.3

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

Tap for more steps...

Step 7.3.1

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

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

Step 7.3.2

Solve for $x$ .

Tap for more steps...

Step 7.3.2.1

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

Tap for more steps...

Step 7.3.2.1.1

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

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

Step 7.3.2.1.2

Simplify the left side.

Tap for more steps...

Step 7.3.2.1.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.3.2.1.2.1.1

Cancel the common factor.

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

Step 7.3.2.1.2.1.2

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

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

Step 7.3.2.1.3

Simplify the right side.

Tap for more steps...

Step 7.3.2.1.3.1

Divide $0$ by $2$ .

$10 x + 7 \geq 0$

Step 7.3.2.2

Subtract $7$ from both sides of the inequality.

$10 x \geq - 7$

Step 7.3.2.3

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

Tap for more steps...

Step 7.3.2.3.1

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

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

Step 7.3.2.3.2

Simplify the left side.

Tap for more steps...

Step 7.3.2.3.2.1

Cancel the common factor of $10$ .

Tap for more steps...

Step 7.3.2.3.2.1.1

Cancel the common factor.

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

Step 7.3.2.3.2.1.2

Divide $x$ by $1$ .

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

Step 7.3.2.3.3

Simplify the right side.

Tap for more steps...

Step 7.3.2.3.3.1

Move the negative in front of the fraction.

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

Step 7.3.3

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

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

Step 7.4

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

Tap for more steps...

Step 7.4.1

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.

$(- \infty, \infty)$

Step 7.5

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