Calculus Examples

$f (x) = \frac{1}{3} + 2 x^{2} - 12 x - 20$

Step 1

Write $f (x) = \frac{1}{3} + 2 x^{2} - 12 x - 20$ as an equation.

$y = \frac{1}{3} + 2 x^{2} - 12 x - 20$

Step 2

Interchange the variables.

$x = \frac{1}{3} + 2 y^{2} - 12 y - 20$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{1}{3} + 2 y^{2} - 12 y - 20 = x$ .

$\frac{1}{3} + 2 y^{2} - 12 y - 20 = x$

Step 3.2

Simplify $\frac{1}{3} + 2 y^{2} - 12 y - 20$ .

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Step 3.2.1

To write $- 20$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$2 y^{2} - 12 y + \frac{1}{3} - 20 \cdot \frac{3}{3} = x$

Step 3.2.2

Combine $- 20$ and $\frac{3}{3}$ .

$2 y^{2} - 12 y + \frac{1}{3} + \frac{- 20 \cdot 3}{3} = x$

Step 3.2.3

Combine the numerators over the common denominator.

$2 y^{2} - 12 y + \frac{1 - 20 \cdot 3}{3} = x$

Step 3.2.4

Simplify the numerator.

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Step 3.2.4.1

Multiply $- 20$ by $3$ .

$2 y^{2} - 12 y + \frac{1 - 60}{3} = x$

Step 3.2.4.2

Subtract $60$ from $1$ .

$2 y^{2} - 12 y + \frac{- 59}{3} = x$

Step 3.2.5

Move the negative in front of the fraction.

$2 y^{2} - 12 y - \frac{59}{3} = x$

Step 3.3

Subtract $x$ from both sides of the equation.

$2 y^{2} - 12 y - \frac{59}{3} - x = 0$

Step 3.4

Multiply through by the least common denominator $3$ , then simplify.

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Step 3.4.1

Apply the distributive property.

$3 (2 y^{2}) + 3 (- 12 y) + 3 (- \frac{59}{3}) + 3 (- x) = 0$

Step 3.4.2

Simplify.

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Step 3.4.2.1

Multiply $2$ by $3$ .

$6 y^{2} + 3 (- 12 y) + 3 (- \frac{59}{3}) + 3 (- x) = 0$

Step 3.4.2.2

Multiply $- 12$ by $3$ .

$6 y^{2} - 36 y + 3 (- \frac{59}{3}) + 3 (- x) = 0$

Step 3.4.2.3

Cancel the common factor of $3$ .

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Step 3.4.2.3.1

Move the leading negative in $- \frac{59}{3}$ into the numerator.

$6 y^{2} - 36 y + 3 (\frac{- 59}{3}) + 3 (- x) = 0$

Step 3.4.2.3.2

Cancel the common factor.

$6 y^{2} - 36 y + 3 (\frac{- 59}{3}) + 3 (- x) = 0$

Step 3.4.2.3.3

Rewrite the expression.

$6 y^{2} - 36 y - 59 + 3 (- x) = 0$

Step 3.4.2.4

Multiply $- 1$ by $3$ .

$6 y^{2} - 36 y - 59 - 3 x = 0$

Step 3.4.3

Move $- 59$ .

$6 y^{2} - 36 y - 3 x - 59 = 0$

Step 3.4.4

Move $- 36 y$ .

$6 y^{2} - 3 x - 36 y - 59 = 0$

Step 3.5

Use the quadratic formula to find the solutions.

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

Step 3.6

Substitute the values $a = 6$ , $b = - 36$ , and $c = - 3 x - 59$ into the quadratic formula and solve for $y$ .

$\frac{36 \pm \sqrt{{(- 36)}^{2} - 4 \cdot (6 \cdot (- 3 x - 59))}}{2 \cdot 6}$

Step 3.7

Simplify.

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Step 3.7.1

Simplify the numerator.

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Step 3.7.1.1

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

$y = \frac{36 \pm \sqrt{1296 - 4 \cdot 6 \cdot (- 3 x - 59)}}{2 \cdot 6}$

Step 3.7.1.2

Multiply $- 4$ by $6$ .

$y = \frac{36 \pm \sqrt{1296 - 24 \cdot (- 3 x - 59)}}{2 \cdot 6}$

Step 3.7.1.3

Apply the distributive property.

$y = \frac{36 \pm \sqrt{1296 - 24 (- 3 x) - 24 \cdot - 59}}{2 \cdot 6}$

Step 3.7.1.4

Multiply $- 3$ by $- 24$ .

$y = \frac{36 \pm \sqrt{1296 + 72 x - 24 \cdot - 59}}{2 \cdot 6}$

Step 3.7.1.5

Multiply $- 24$ by $- 59$ .

$y = \frac{36 \pm \sqrt{1296 + 72 x + 1416}}{2 \cdot 6}$

Step 3.7.1.6

Add $1296$ and $1416$ .

$y = \frac{36 \pm \sqrt{72 x + 2712}}{2 \cdot 6}$

Step 3.7.1.7

Factor $24$ out of $72 x + 2712$ .

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Step 3.7.1.7.1

Factor $24$ out of $72 x$ .

$y = \frac{36 \pm \sqrt{24 (3 x) + 2712}}{2 \cdot 6}$

Step 3.7.1.7.2

Factor $24$ out of $2712$ .

$y = \frac{36 \pm \sqrt{24 (3 x) + 24 (113)}}{2 \cdot 6}$

Step 3.7.1.7.3

Factor $24$ out of $24 (3 x) + 24 (113)$ .

$y = \frac{36 \pm \sqrt{24 (3 x + 113)}}{2 \cdot 6}$

Step 3.7.1.8

Rewrite $24 (3 x + 113)$ as $2^{2} \cdot (6 (3 x + 113))$ .

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Step 3.7.1.8.1

Factor $4$ out of $24$ .

$y = \frac{36 \pm \sqrt{4 (6) (3 x + 113)}}{2 \cdot 6}$

Step 3.7.1.8.2

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

$y = \frac{36 \pm \sqrt{2^{2} \cdot (6 (3 x + 113))}}{2 \cdot 6}$

Step 3.7.1.8.3

Add parentheses.

$y = \frac{36 \pm \sqrt{2^{2} \cdot (6 (3 x + 113))}}{2 \cdot 6}$

Step 3.7.1.9

Pull terms out from under the radical.

$y = \frac{36 \pm 2 \sqrt{6 (3 x + 113)}}{2 \cdot 6}$

Step 3.7.2

Multiply $2$ by $6$ .

$y = \frac{36 \pm 2 \sqrt{6 (3 x + 113)}}{12}$

Step 3.7.3

Simplify $\frac{36 \pm 2 \sqrt{6 (3 x + 113)}}{12}$ .

$y = \frac{18 \pm \sqrt{6 (3 x + 113)}}{6}$

Step 3.8

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

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Step 3.8.1

Simplify the numerator.

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Step 3.8.1.1

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

$y = \frac{36 \pm \sqrt{1296 - 4 \cdot 6 \cdot (- 3 x - 59)}}{2 \cdot 6}$

Step 3.8.1.2

Multiply $- 4$ by $6$ .

$y = \frac{36 \pm \sqrt{1296 - 24 \cdot (- 3 x - 59)}}{2 \cdot 6}$

Step 3.8.1.3

Apply the distributive property.

$y = \frac{36 \pm \sqrt{1296 - 24 (- 3 x) - 24 \cdot - 59}}{2 \cdot 6}$

Step 3.8.1.4

Multiply $- 3$ by $- 24$ .

$y = \frac{36 \pm \sqrt{1296 + 72 x - 24 \cdot - 59}}{2 \cdot 6}$

Step 3.8.1.5

Multiply $- 24$ by $- 59$ .

$y = \frac{36 \pm \sqrt{1296 + 72 x + 1416}}{2 \cdot 6}$

Step 3.8.1.6

Add $1296$ and $1416$ .

$y = \frac{36 \pm \sqrt{72 x + 2712}}{2 \cdot 6}$

Step 3.8.1.7

Factor $24$ out of $72 x + 2712$ .

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Step 3.8.1.7.1

Factor $24$ out of $72 x$ .

$y = \frac{36 \pm \sqrt{24 (3 x) + 2712}}{2 \cdot 6}$

Step 3.8.1.7.2

Factor $24$ out of $2712$ .

$y = \frac{36 \pm \sqrt{24 (3 x) + 24 (113)}}{2 \cdot 6}$

Step 3.8.1.7.3

Factor $24$ out of $24 (3 x) + 24 (113)$ .

$y = \frac{36 \pm \sqrt{24 (3 x + 113)}}{2 \cdot 6}$

Step 3.8.1.8

Rewrite $24 (3 x + 113)$ as $2^{2} \cdot (6 (3 x + 113))$ .

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Step 3.8.1.8.1

Factor $4$ out of $24$ .

$y = \frac{36 \pm \sqrt{4 (6) (3 x + 113)}}{2 \cdot 6}$

Step 3.8.1.8.2

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

$y = \frac{36 \pm \sqrt{2^{2} \cdot (6 (3 x + 113))}}{2 \cdot 6}$

Step 3.8.1.8.3

Add parentheses.

$y = \frac{36 \pm \sqrt{2^{2} \cdot (6 (3 x + 113))}}{2 \cdot 6}$

Step 3.8.1.9

Pull terms out from under the radical.

$y = \frac{36 \pm 2 \sqrt{6 (3 x + 113)}}{2 \cdot 6}$

Step 3.8.2

Multiply $2$ by $6$ .

$y = \frac{36 \pm 2 \sqrt{6 (3 x + 113)}}{12}$

Step 3.8.3

Simplify $\frac{36 \pm 2 \sqrt{6 (3 x + 113)}}{12}$ .

$y = \frac{18 \pm \sqrt{6 (3 x + 113)}}{6}$

Step 3.8.4

Change the $\pm$ to $+$ .

$y = \frac{18 + \sqrt{6 (3 x + 113)}}{6}$

Step 3.9

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

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Step 3.9.1

Simplify the numerator.

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Step 3.9.1.1

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

$y = \frac{36 \pm \sqrt{1296 - 4 \cdot 6 \cdot (- 3 x - 59)}}{2 \cdot 6}$

Step 3.9.1.2

Multiply $- 4$ by $6$ .

$y = \frac{36 \pm \sqrt{1296 - 24 \cdot (- 3 x - 59)}}{2 \cdot 6}$

Step 3.9.1.3

Apply the distributive property.

$y = \frac{36 \pm \sqrt{1296 - 24 (- 3 x) - 24 \cdot - 59}}{2 \cdot 6}$

Step 3.9.1.4

Multiply $- 3$ by $- 24$ .

$y = \frac{36 \pm \sqrt{1296 + 72 x - 24 \cdot - 59}}{2 \cdot 6}$

Step 3.9.1.5

Multiply $- 24$ by $- 59$ .

$y = \frac{36 \pm \sqrt{1296 + 72 x + 1416}}{2 \cdot 6}$

Step 3.9.1.6

Add $1296$ and $1416$ .

$y = \frac{36 \pm \sqrt{72 x + 2712}}{2 \cdot 6}$

Step 3.9.1.7

Factor $24$ out of $72 x + 2712$ .

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Step 3.9.1.7.1

Factor $24$ out of $72 x$ .

$y = \frac{36 \pm \sqrt{24 (3 x) + 2712}}{2 \cdot 6}$

Step 3.9.1.7.2

Factor $24$ out of $2712$ .

$y = \frac{36 \pm \sqrt{24 (3 x) + 24 (113)}}{2 \cdot 6}$

Step 3.9.1.7.3

Factor $24$ out of $24 (3 x) + 24 (113)$ .

$y = \frac{36 \pm \sqrt{24 (3 x + 113)}}{2 \cdot 6}$

Step 3.9.1.8

Rewrite $24 (3 x + 113)$ as $2^{2} \cdot (6 (3 x + 113))$ .

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Step 3.9.1.8.1

Factor $4$ out of $24$ .

$y = \frac{36 \pm \sqrt{4 (6) (3 x + 113)}}{2 \cdot 6}$

Step 3.9.1.8.2

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

$y = \frac{36 \pm \sqrt{2^{2} \cdot (6 (3 x + 113))}}{2 \cdot 6}$

Step 3.9.1.8.3

Add parentheses.

$y = \frac{36 \pm \sqrt{2^{2} \cdot (6 (3 x + 113))}}{2 \cdot 6}$

Step 3.9.1.9

Pull terms out from under the radical.

$y = \frac{36 \pm 2 \sqrt{6 (3 x + 113)}}{2 \cdot 6}$

Step 3.9.2

Multiply $2$ by $6$ .

$y = \frac{36 \pm 2 \sqrt{6 (3 x + 113)}}{12}$

Step 3.9.3

Simplify $\frac{36 \pm 2 \sqrt{6 (3 x + 113)}}{12}$ .

$y = \frac{18 \pm \sqrt{6 (3 x + 113)}}{6}$

Step 3.9.4

Change the $\pm$ to $-$ .

$y = \frac{18 - \sqrt{6 (3 x + 113)}}{6}$

Step 3.10

The final answer is the combination of both solutions.

$y = \frac{18 + \sqrt{6 (3 x + 113)}}{6}$

$y = \frac{18 - \sqrt{6 (3 x + 113)}}{6}$

$y = \frac{18 + \sqrt{6 (3 x + 113)}}{6}$

$y = \frac{18 - \sqrt{6 (3 x + 113)}}{6}$

Step 4

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

$f^{- 1} (x) = \frac{18 + \sqrt{6 (3 x + 113)}}{6}, \frac{18 - \sqrt{6 (3 x + 113)}}{6}$

Step 5

Verify if $f^{- 1} (x) = \frac{18 + \sqrt{6 (3 x + 113)}}{6}, \frac{18 - \sqrt{6 (3 x + 113)}}{6}$ is the inverse of $f (x) = \frac{1}{3} + 2 x^{2} - 12 x - 20$ .

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Step 5.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = \frac{1}{3} + 2 x^{2} - 12 x - 20$ and $f^{- 1} (x) = \frac{18 + \sqrt{6 (3 x + 113)}}{6}, \frac{18 - \sqrt{6 (3 x + 113)}}{6}$ and compare them.

Step 5.2

Find the range of $f (x) = \frac{1}{3} + 2 x^{2} - 12 x - 20$ .

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Step 5.2.1

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

Interval Notation:

$[- \frac{113}{3}, \infty)$

Step 5.3

Find the domain of $\frac{18 + \sqrt{6 (3 x + 113)}}{6}$ .

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Step 5.3.1

Set the radicand in $\sqrt{6 (3 x + 113)}$ greater than or equal to $0$ to find where the expression is defined.

$6 (3 x + 113) \geq 0$

Step 5.3.2

Solve for $x$ .

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Step 5.3.2.1

Divide each term in $6 (3 x + 113) \geq 0$ by $6$ and simplify.

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Step 5.3.2.1.1

Divide each term in $6 (3 x + 113) \geq 0$ by $6$ .

$\frac{6 (3 x + 113)}{6} \geq \frac{0}{6}$

Step 5.3.2.1.2

Simplify the left side.

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Step 5.3.2.1.2.1

Cancel the common factor of $6$ .

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Step 5.3.2.1.2.1.1

Cancel the common factor.

$\frac{6 (3 x + 113)}{6} \geq \frac{0}{6}$

Step 5.3.2.1.2.1.2

Divide $3 x + 113$ by $1$ .

$3 x + 113 \geq \frac{0}{6}$

Step 5.3.2.1.3

Simplify the right side.

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Step 5.3.2.1.3.1

Divide $0$ by $6$ .

$3 x + 113 \geq 0$

Step 5.3.2.2

Subtract $113$ from both sides of the inequality.

$3 x \geq - 113$

Step 5.3.2.3

Divide each term in $3 x \geq - 113$ by $3$ and simplify.

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Step 5.3.2.3.1

Divide each term in $3 x \geq - 113$ by $3$ .

$\frac{3 x}{3} \geq \frac{- 113}{3}$

Step 5.3.2.3.2

Simplify the left side.

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Step 5.3.2.3.2.1

Cancel the common factor of $3$ .

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Step 5.3.2.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{- 113}{3}$

Step 5.3.2.3.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 113}{3}$

Step 5.3.2.3.3

Simplify the right side.

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Step 5.3.2.3.3.1

Move the negative in front of the fraction.

$x \geq - \frac{113}{3}$

Step 5.3.3

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

$[- \frac{113}{3}, \infty)$

Step 5.4

Find the domain of $f (x) = \frac{1}{3} + 2 x^{2} - 12 x - 20$ .

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Step 5.4.1

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 5.5

Since the domain of $f^{- 1} (x) = \frac{18 + \sqrt{6 (3 x + 113)}}{6}, \frac{18 - \sqrt{6 (3 x + 113)}}{6}$ is the range of $f (x) = \frac{1}{3} + 2 x^{2} - 12 x - 20$ and the range of $f^{- 1} (x) = \frac{18 + \sqrt{6 (3 x + 113)}}{6}, \frac{18 - \sqrt{6 (3 x + 113)}}{6}$ is the domain of $f (x) = \frac{1}{3} + 2 x^{2} - 12 x - 20$ , then $f^{- 1} (x) = \frac{18 + \sqrt{6 (3 x + 113)}}{6}, \frac{18 - \sqrt{6 (3 x + 113)}}{6}$ is the inverse of $f (x) = \frac{1}{3} + 2 x^{2} - 12 x - 20$ .

$f^{- 1} (x) = \frac{18 + \sqrt{6 (3 x + 113)}}{6}, \frac{18 - \sqrt{6 (3 x + 113)}}{6}$

Step 6