Algebra Examples

$y = \frac{1}{6} x^{2} - \frac{4}{3} x + \frac{43}{9}$

Step 1

Interchange the variables.

$x = \frac{1}{6} y^{2} - \frac{4}{3} y + \frac{43}{9}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{1}{6} y^{2} - \frac{4}{3} y + \frac{43}{9} = x$ .

$\frac{1}{6} y^{2} - \frac{4}{3} y + \frac{43}{9} = x$

Step 2.2

Simplify each term.

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Step 2.2.1

Combine $\frac{1}{6}$ and $y^{2}$ .

$\frac{y^{2}}{6} - \frac{4}{3} y + \frac{43}{9} = x$

Step 2.2.2

Combine $y$ and $\frac{4}{3}$ .

$\frac{y^{2}}{6} - \frac{y \cdot 4}{3} + \frac{43}{9} = x$

Step 2.2.3

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

$\frac{y^{2}}{6} - \frac{4 y}{3} + \frac{43}{9} = x$

Step 2.3

Subtract $x$ from both sides of the equation.

$\frac{y^{2}}{6} - \frac{4 y}{3} + \frac{43}{9} - x = 0$

Step 2.4

Multiply through by the least common denominator $18$ , then simplify.

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Step 2.4.1

Apply the distributive property.

$18 (\frac{y^{2}}{6}) + 18 (- \frac{4 y}{3}) + 18 (\frac{43}{9}) + 18 (- x) = 0$

Step 2.4.2

Simplify.

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Step 2.4.2.1

Cancel the common factor of $6$ .

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Step 2.4.2.1.1

Factor $6$ out of $18$ .

$6 (3) (\frac{y^{2}}{6}) + 18 (- \frac{4 y}{3}) + 18 (\frac{43}{9}) + 18 (- x) = 0$

Step 2.4.2.1.2

Cancel the common factor.

$6 \cdot (3 (\frac{y^{2}}{6})) + 18 (- \frac{4 y}{3}) + 18 (\frac{43}{9}) + 18 (- x) = 0$

Step 2.4.2.1.3

Rewrite the expression.

$3 y^{2} + 18 (- \frac{4 y}{3}) + 18 (\frac{43}{9}) + 18 (- x) = 0$

Step 2.4.2.2

Cancel the common factor of $3$ .

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Step 2.4.2.2.1

Move the leading negative in $- \frac{4 y}{3}$ into the numerator.

$3 y^{2} + 18 (\frac{- 4 y}{3}) + 18 (\frac{43}{9}) + 18 (- x) = 0$

Step 2.4.2.2.2

Factor $3$ out of $18$ .

$3 y^{2} + 3 (6) (\frac{- 4 y}{3}) + 18 (\frac{43}{9}) + 18 (- x) = 0$

Step 2.4.2.2.3

Cancel the common factor.

$3 y^{2} + 3 \cdot (6 (\frac{- 4 y}{3})) + 18 (\frac{43}{9}) + 18 (- x) = 0$

Step 2.4.2.2.4

Rewrite the expression.

$3 y^{2} + 6 (- 4 y) + 18 (\frac{43}{9}) + 18 (- x) = 0$

Step 2.4.2.3

Multiply $- 4$ by $6$ .

$3 y^{2} - 24 y + 18 (\frac{43}{9}) + 18 (- x) = 0$

Step 2.4.2.4

Cancel the common factor of $9$ .

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Step 2.4.2.4.1

Factor $9$ out of $18$ .

$3 y^{2} - 24 y + 9 (2) (\frac{43}{9}) + 18 (- x) = 0$

Step 2.4.2.4.2

Cancel the common factor.

$3 y^{2} - 24 y + 9 \cdot (2 (\frac{43}{9})) + 18 (- x) = 0$

Step 2.4.2.4.3

Rewrite the expression.

$3 y^{2} - 24 y + 2 \cdot 43 + 18 (- x) = 0$

Step 2.4.2.5

Multiply $2$ by $43$ .

$3 y^{2} - 24 y + 86 + 18 (- x) = 0$

Step 2.4.2.6

Multiply $- 1$ by $18$ .

$3 y^{2} - 24 y + 86 - 18 x = 0$

Step 2.4.3

Move $86$ .

$3 y^{2} - 24 y - 18 x + 86 = 0$

Step 2.4.4

Move $- 24 y$ .

$3 y^{2} - 18 x - 24 y + 86 = 0$

Step 2.5

Use the quadratic formula to find the solutions.

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

Step 2.6

Substitute the values $a = 3$ , $b = - 24$ , and $c = - 18 x + 86$ into the quadratic formula and solve for $y$ .

$\frac{24 \pm \sqrt{{(- 24)}^{2} - 4 \cdot (3 \cdot (- 18 x + 86))}}{2 \cdot 3}$

Step 2.7

Simplify.

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

Raise $- 24$ to the power of $2$ .

$y = \frac{24 \pm \sqrt{576 - 4 \cdot 3 \cdot (- 18 x + 86)}}{2 \cdot 3}$

Step 2.7.1.2

Multiply $- 4$ by $3$ .

$y = \frac{24 \pm \sqrt{576 - 12 \cdot (- 18 x + 86)}}{2 \cdot 3}$

Step 2.7.1.3

Apply the distributive property.

$y = \frac{24 \pm \sqrt{576 - 12 (- 18 x) - 12 \cdot 86}}{2 \cdot 3}$

Step 2.7.1.4

Multiply $- 18$ by $- 12$ .

$y = \frac{24 \pm \sqrt{576 + 216 x - 12 \cdot 86}}{2 \cdot 3}$

Step 2.7.1.5

Multiply $- 12$ by $86$ .

$y = \frac{24 \pm \sqrt{576 + 216 x - 1032}}{2 \cdot 3}$

Step 2.7.1.6

Subtract $1032$ from $576$ .

$y = \frac{24 \pm \sqrt{216 x - 456}}{2 \cdot 3}$

Step 2.7.1.7

Factor $24$ out of $216 x - 456$ .

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Step 2.7.1.7.1

Factor $24$ out of $216 x$ .

$y = \frac{24 \pm \sqrt{24 (9 x) - 456}}{2 \cdot 3}$

Step 2.7.1.7.2

Factor $24$ out of $- 456$ .

$y = \frac{24 \pm \sqrt{24 (9 x) + 24 (- 19)}}{2 \cdot 3}$

Step 2.7.1.7.3

Factor $24$ out of $24 (9 x) + 24 (- 19)$ .

$y = \frac{24 \pm \sqrt{24 (9 x - 19)}}{2 \cdot 3}$

Step 2.7.1.8

Rewrite $24 (9 x - 19)$ as $2^{2} \cdot (6 (3^{2} x - 19))$ .

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Step 2.7.1.8.1

Factor $4$ out of $24$ .

$y = \frac{24 \pm \sqrt{4 (6) (9 x - 19)}}{2 \cdot 3}$

Step 2.7.1.8.2

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

$y = \frac{24 \pm \sqrt{2^{2} \cdot (6 (9 x - 19))}}{2 \cdot 3}$

Step 2.7.1.8.3

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

$y = \frac{24 \pm \sqrt{2^{2} \cdot (6 (3^{2} x - 19))}}{2 \cdot 3}$

Step 2.7.1.8.4

Add parentheses.

$y = \frac{24 \pm \sqrt{2^{2} \cdot (6 (3^{2} x - 19))}}{2 \cdot 3}$

Step 2.7.1.9

Pull terms out from under the radical.

$y = \frac{24 \pm 2 \sqrt{6 (3^{2} x - 19)}}{2 \cdot 3}$

Step 2.7.1.10

Raise $3$ to the power of $2$ .

$y = \frac{24 \pm 2 \sqrt{6 (9 x - 19)}}{2 \cdot 3}$

Step 2.7.2

Multiply $2$ by $3$ .

$y = \frac{24 \pm 2 \sqrt{6 (9 x - 19)}}{6}$

Step 2.7.3

Simplify $\frac{24 \pm 2 \sqrt{6 (9 x - 19)}}{6}$ .

$y = \frac{12 \pm \sqrt{6 (9 x - 19)}}{3}$

Step 2.8

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

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Step 2.8.1

Simplify the numerator.

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Step 2.8.1.1

Raise $- 24$ to the power of $2$ .

$y = \frac{24 \pm \sqrt{576 - 4 \cdot 3 \cdot (- 18 x + 86)}}{2 \cdot 3}$

Step 2.8.1.2

Multiply $- 4$ by $3$ .

$y = \frac{24 \pm \sqrt{576 - 12 \cdot (- 18 x + 86)}}{2 \cdot 3}$

Step 2.8.1.3

Apply the distributive property.

$y = \frac{24 \pm \sqrt{576 - 12 (- 18 x) - 12 \cdot 86}}{2 \cdot 3}$

Step 2.8.1.4

Multiply $- 18$ by $- 12$ .

$y = \frac{24 \pm \sqrt{576 + 216 x - 12 \cdot 86}}{2 \cdot 3}$

Step 2.8.1.5

Multiply $- 12$ by $86$ .

$y = \frac{24 \pm \sqrt{576 + 216 x - 1032}}{2 \cdot 3}$

Step 2.8.1.6

Subtract $1032$ from $576$ .

$y = \frac{24 \pm \sqrt{216 x - 456}}{2 \cdot 3}$

Step 2.8.1.7

Factor $24$ out of $216 x - 456$ .

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Step 2.8.1.7.1

Factor $24$ out of $216 x$ .

$y = \frac{24 \pm \sqrt{24 (9 x) - 456}}{2 \cdot 3}$

Step 2.8.1.7.2

Factor $24$ out of $- 456$ .

$y = \frac{24 \pm \sqrt{24 (9 x) + 24 (- 19)}}{2 \cdot 3}$

Step 2.8.1.7.3

Factor $24$ out of $24 (9 x) + 24 (- 19)$ .

$y = \frac{24 \pm \sqrt{24 (9 x - 19)}}{2 \cdot 3}$

Step 2.8.1.8

Rewrite $24 (9 x - 19)$ as $2^{2} \cdot (6 (3^{2} x - 19))$ .

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Step 2.8.1.8.1

Factor $4$ out of $24$ .

$y = \frac{24 \pm \sqrt{4 (6) (9 x - 19)}}{2 \cdot 3}$

Step 2.8.1.8.2

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

$y = \frac{24 \pm \sqrt{2^{2} \cdot (6 (9 x - 19))}}{2 \cdot 3}$

Step 2.8.1.8.3

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

$y = \frac{24 \pm \sqrt{2^{2} \cdot (6 (3^{2} x - 19))}}{2 \cdot 3}$

Step 2.8.1.8.4

Add parentheses.

$y = \frac{24 \pm \sqrt{2^{2} \cdot (6 (3^{2} x - 19))}}{2 \cdot 3}$

Step 2.8.1.9

Pull terms out from under the radical.

$y = \frac{24 \pm 2 \sqrt{6 (3^{2} x - 19)}}{2 \cdot 3}$

Step 2.8.1.10

Raise $3$ to the power of $2$ .

$y = \frac{24 \pm 2 \sqrt{6 (9 x - 19)}}{2 \cdot 3}$

Step 2.8.2

Multiply $2$ by $3$ .

$y = \frac{24 \pm 2 \sqrt{6 (9 x - 19)}}{6}$

Step 2.8.3

Simplify $\frac{24 \pm 2 \sqrt{6 (9 x - 19)}}{6}$ .

$y = \frac{12 \pm \sqrt{6 (9 x - 19)}}{3}$

Step 2.8.4

Change the $\pm$ to $+$ .

$y = \frac{12 + \sqrt{6 (9 x - 19)}}{3}$

Step 2.9

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

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Step 2.9.1

Simplify the numerator.

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Step 2.9.1.1

Raise $- 24$ to the power of $2$ .

$y = \frac{24 \pm \sqrt{576 - 4 \cdot 3 \cdot (- 18 x + 86)}}{2 \cdot 3}$

Step 2.9.1.2

Multiply $- 4$ by $3$ .

$y = \frac{24 \pm \sqrt{576 - 12 \cdot (- 18 x + 86)}}{2 \cdot 3}$

Step 2.9.1.3

Apply the distributive property.

$y = \frac{24 \pm \sqrt{576 - 12 (- 18 x) - 12 \cdot 86}}{2 \cdot 3}$

Step 2.9.1.4

Multiply $- 18$ by $- 12$ .

$y = \frac{24 \pm \sqrt{576 + 216 x - 12 \cdot 86}}{2 \cdot 3}$

Step 2.9.1.5

Multiply $- 12$ by $86$ .

$y = \frac{24 \pm \sqrt{576 + 216 x - 1032}}{2 \cdot 3}$

Step 2.9.1.6

Subtract $1032$ from $576$ .

$y = \frac{24 \pm \sqrt{216 x - 456}}{2 \cdot 3}$

Step 2.9.1.7

Factor $24$ out of $216 x - 456$ .

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Step 2.9.1.7.1

Factor $24$ out of $216 x$ .

$y = \frac{24 \pm \sqrt{24 (9 x) - 456}}{2 \cdot 3}$

Step 2.9.1.7.2

Factor $24$ out of $- 456$ .

$y = \frac{24 \pm \sqrt{24 (9 x) + 24 (- 19)}}{2 \cdot 3}$

Step 2.9.1.7.3

Factor $24$ out of $24 (9 x) + 24 (- 19)$ .

$y = \frac{24 \pm \sqrt{24 (9 x - 19)}}{2 \cdot 3}$

Step 2.9.1.8

Rewrite $24 (9 x - 19)$ as $2^{2} \cdot (6 (3^{2} x - 19))$ .

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Step 2.9.1.8.1

Factor $4$ out of $24$ .

$y = \frac{24 \pm \sqrt{4 (6) (9 x - 19)}}{2 \cdot 3}$

Step 2.9.1.8.2

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

$y = \frac{24 \pm \sqrt{2^{2} \cdot (6 (9 x - 19))}}{2 \cdot 3}$

Step 2.9.1.8.3

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

$y = \frac{24 \pm \sqrt{2^{2} \cdot (6 (3^{2} x - 19))}}{2 \cdot 3}$

Step 2.9.1.8.4

Add parentheses.

$y = \frac{24 \pm \sqrt{2^{2} \cdot (6 (3^{2} x - 19))}}{2 \cdot 3}$

Step 2.9.1.9

Pull terms out from under the radical.

$y = \frac{24 \pm 2 \sqrt{6 (3^{2} x - 19)}}{2 \cdot 3}$

Step 2.9.1.10

Raise $3$ to the power of $2$ .

$y = \frac{24 \pm 2 \sqrt{6 (9 x - 19)}}{2 \cdot 3}$

Step 2.9.2

Multiply $2$ by $3$ .

$y = \frac{24 \pm 2 \sqrt{6 (9 x - 19)}}{6}$

Step 2.9.3

Simplify $\frac{24 \pm 2 \sqrt{6 (9 x - 19)}}{6}$ .

$y = \frac{12 \pm \sqrt{6 (9 x - 19)}}{3}$

Step 2.9.4

Change the $\pm$ to $-$ .

$y = \frac{12 - \sqrt{6 (9 x - 19)}}{3}$

Step 2.10

The final answer is the combination of both solutions.

$y = \frac{12 + \sqrt{6 (9 x - 19)}}{3}$

$y = \frac{12 - \sqrt{6 (9 x - 19)}}{3}$

$y = \frac{12 + \sqrt{6 (9 x - 19)}}{3}$

$y = \frac{12 - \sqrt{6 (9 x - 19)}}{3}$

Step 3

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

$f^{- 1} (x) = \frac{12 + \sqrt{6 (9 x - 19)}}{3}, \frac{12 - \sqrt{6 (9 x - 19)}}{3}$

Step 4

Verify if $f^{- 1} (x) = \frac{12 + \sqrt{6 (9 x - 19)}}{3}, \frac{12 - \sqrt{6 (9 x - 19)}}{3}$ is the inverse of $f (x) = \frac{1}{6} x^{2} - \frac{4}{3} x + \frac{43}{9}$ .

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Step 4.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = \frac{1}{6} x^{2} - \frac{4}{3} x + \frac{43}{9}$ and $f^{- 1} (x) = \frac{12 + \sqrt{6 (9 x - 19)}}{3}, \frac{12 - \sqrt{6 (9 x - 19)}}{3}$ and compare them.

Step 4.2

Find the range of $f (x) = \frac{1}{6} x^{2} - \frac{4}{3} x + \frac{43}{9}$ .

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Step 4.2.1

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

Interval Notation:

$[\frac{19}{9}, \infty)$

Step 4.3

Find the domain of $\frac{12 + \sqrt{6 (9 x - 19)}}{3}$ .

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Step 4.3.1

Set the radicand in $\sqrt{6 (9 x - 19)}$ greater than or equal to $0$ to find where the expression is defined.

$6 (9 x - 19) \geq 0$

Step 4.3.2

Solve for $x$ .

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Step 4.3.2.1

Divide each term in $6 (9 x - 19) \geq 0$ by $6$ and simplify.

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Step 4.3.2.1.1

Divide each term in $6 (9 x - 19) \geq 0$ by $6$ .

$\frac{6 (9 x - 19)}{6} \geq \frac{0}{6}$

Step 4.3.2.1.2

Simplify the left side.

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Step 4.3.2.1.2.1

Cancel the common factor of $6$ .

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Step 4.3.2.1.2.1.1

Cancel the common factor.

$\frac{6 (9 x - 19)}{6} \geq \frac{0}{6}$

Step 4.3.2.1.2.1.2

Divide $9 x - 19$ by $1$ .

$9 x - 19 \geq \frac{0}{6}$

Step 4.3.2.1.3

Simplify the right side.

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Step 4.3.2.1.3.1

Divide $0$ by $6$ .

$9 x - 19 \geq 0$

Step 4.3.2.2

Add $19$ to both sides of the inequality.

$9 x \geq 19$

Step 4.3.2.3

Divide each term in $9 x \geq 19$ by $9$ and simplify.

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Step 4.3.2.3.1

Divide each term in $9 x \geq 19$ by $9$ .

$\frac{9 x}{9} \geq \frac{19}{9}$

Step 4.3.2.3.2

Simplify the left side.

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Step 4.3.2.3.2.1

Cancel the common factor of $9$ .

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Step 4.3.2.3.2.1.1

Cancel the common factor.

$\frac{9 x}{9} \geq \frac{19}{9}$

Step 4.3.2.3.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{19}{9}$

Step 4.3.3

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

$[\frac{19}{9}, \infty)$

Step 4.4

Find the domain of $f (x) = \frac{1}{6} x^{2} - \frac{4}{3} x + \frac{43}{9}$ .

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Step 4.4.1

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 4.5

Since the domain of $f^{- 1} (x) = \frac{12 + \sqrt{6 (9 x - 19)}}{3}, \frac{12 - \sqrt{6 (9 x - 19)}}{3}$ is the range of $f (x) = \frac{1}{6} x^{2} - \frac{4}{3} x + \frac{43}{9}$ and the range of $f^{- 1} (x) = \frac{12 + \sqrt{6 (9 x - 19)}}{3}, \frac{12 - \sqrt{6 (9 x - 19)}}{3}$ is the domain of $f (x) = \frac{1}{6} x^{2} - \frac{4}{3} x + \frac{43}{9}$ , then $f^{- 1} (x) = \frac{12 + \sqrt{6 (9 x - 19)}}{3}, \frac{12 - \sqrt{6 (9 x - 19)}}{3}$ is the inverse of $f (x) = \frac{1}{6} x^{2} - \frac{4}{3} x + \frac{43}{9}$ .

$f^{- 1} (x) = \frac{12 + \sqrt{6 (9 x - 19)}}{3}, \frac{12 - \sqrt{6 (9 x - 19)}}{3}$

Step 5