Finite Math Examples

$f (x) = - 3 x^{2} - 6 x + 2$

Step 1

Write $f (x) = - 3 x^{2} - 6 x + 2$ as an equation.

$y = - 3 x^{2} - 6 x + 2$

Step 2

Interchange the variables.

$x = - 3 y^{2} - 6 y + 2$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $- 3 y^{2} - 6 y + 2 = x$ .

$- 3 y^{2} - 6 y + 2 = x$

Step 3.2

Subtract $x$ from both sides of the equation.

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

Step 3.3

Use the quadratic formula to find the solutions.

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

Step 3.4

Substitute the values $a = - 3$ , $b = - 6$ , and $c = 2 - x$ into the quadratic formula and solve for $y$ .

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

Step 3.5

Simplify.

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Raise $- 6$ to the power of $2$ .

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

Step 3.5.1.2

Multiply $- 4$ by $- 3$ .

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

Step 3.5.1.3

Apply the distributive property.

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

Step 3.5.1.4

Multiply $12$ by $2$ .

$y = \frac{6 \pm \sqrt{36 + 24 + 12 (- x)}}{2 \cdot - 3}$

Step 3.5.1.5

Multiply $- 1$ by $12$ .

$y = \frac{6 \pm \sqrt{36 + 24 - 12 x}}{2 \cdot - 3}$

Step 3.5.1.6

Add $36$ and $24$ .

$y = \frac{6 \pm \sqrt{60 - 12 x}}{2 \cdot - 3}$

Step 3.5.1.7

Factor $12$ out of $60 - 12 x$ .

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Step 3.5.1.7.1

Factor $12$ out of $60$ .

$y = \frac{6 \pm \sqrt{12 (5) - 12 x}}{2 \cdot - 3}$

Step 3.5.1.7.2

Factor $12$ out of $- 12 x$ .

$y = \frac{6 \pm \sqrt{12 (5) + 12 (- x)}}{2 \cdot - 3}$

Step 3.5.1.7.3

Factor $12$ out of $12 (5) + 12 (- x)$ .

$y = \frac{6 \pm \sqrt{12 (5 - x)}}{2 \cdot - 3}$

Step 3.5.1.8

Rewrite $12 (5 - x)$ as $2^{2} \cdot (3 (5 - x))$ .

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Step 3.5.1.8.1

Factor $4$ out of $12$ .

$y = \frac{6 \pm \sqrt{4 (3) (5 - x)}}{2 \cdot - 3}$

Step 3.5.1.8.2

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

$y = \frac{6 \pm \sqrt{2^{2} \cdot (3 (5 - x))}}{2 \cdot - 3}$

Step 3.5.1.8.3

Add parentheses.

$y = \frac{6 \pm \sqrt{2^{2} \cdot (3 (5 - x))}}{2 \cdot - 3}$

Step 3.5.1.9

Pull terms out from under the radical.

$y = \frac{6 \pm 2 \sqrt{3 (5 - x)}}{2 \cdot - 3}$

Step 3.5.2

Multiply $2$ by $- 3$ .

$y = \frac{6 \pm 2 \sqrt{3 (5 - x)}}{- 6}$

Step 3.5.3

Simplify $\frac{6 \pm 2 \sqrt{3 (5 - x)}}{- 6}$ .

$y = \frac{3 \pm \sqrt{3 (5 - x)}}{- 3}$

Step 3.5.4

Move the negative in front of the fraction.

$y = - \frac{3 \pm \sqrt{3 (5 - x)}}{3}$

Step 3.6

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

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Raise $- 6$ to the power of $2$ .

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

Step 3.6.1.2

Multiply $- 4$ by $- 3$ .

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

Step 3.6.1.3

Apply the distributive property.

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

Step 3.6.1.4

Multiply $12$ by $2$ .

$y = \frac{6 \pm \sqrt{36 + 24 + 12 (- x)}}{2 \cdot - 3}$

Step 3.6.1.5

Multiply $- 1$ by $12$ .

$y = \frac{6 \pm \sqrt{36 + 24 - 12 x}}{2 \cdot - 3}$

Step 3.6.1.6

Add $36$ and $24$ .

$y = \frac{6 \pm \sqrt{60 - 12 x}}{2 \cdot - 3}$

Step 3.6.1.7

Factor $12$ out of $60 - 12 x$ .

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Step 3.6.1.7.1

Factor $12$ out of $60$ .

$y = \frac{6 \pm \sqrt{12 (5) - 12 x}}{2 \cdot - 3}$

Step 3.6.1.7.2

Factor $12$ out of $- 12 x$ .

$y = \frac{6 \pm \sqrt{12 (5) + 12 (- x)}}{2 \cdot - 3}$

Step 3.6.1.7.3

Factor $12$ out of $12 (5) + 12 (- x)$ .

$y = \frac{6 \pm \sqrt{12 (5 - x)}}{2 \cdot - 3}$

Step 3.6.1.8

Rewrite $12 (5 - x)$ as $2^{2} \cdot (3 (5 - x))$ .

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Step 3.6.1.8.1

Factor $4$ out of $12$ .

$y = \frac{6 \pm \sqrt{4 (3) (5 - x)}}{2 \cdot - 3}$

Step 3.6.1.8.2

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

$y = \frac{6 \pm \sqrt{2^{2} \cdot (3 (5 - x))}}{2 \cdot - 3}$

Step 3.6.1.8.3

Add parentheses.

$y = \frac{6 \pm \sqrt{2^{2} \cdot (3 (5 - x))}}{2 \cdot - 3}$

Step 3.6.1.9

Pull terms out from under the radical.

$y = \frac{6 \pm 2 \sqrt{3 (5 - x)}}{2 \cdot - 3}$

Step 3.6.2

Multiply $2$ by $- 3$ .

$y = \frac{6 \pm 2 \sqrt{3 (5 - x)}}{- 6}$

Step 3.6.3

Simplify $\frac{6 \pm 2 \sqrt{3 (5 - x)}}{- 6}$ .

$y = \frac{3 \pm \sqrt{3 (5 - x)}}{- 3}$

Step 3.6.4

Move the negative in front of the fraction.

$y = - \frac{3 \pm \sqrt{3 (5 - x)}}{3}$

Step 3.6.5

Change the $\pm$ to $+$ .

$y = - \frac{3 + \sqrt{3 (5 - x)}}{3}$

Step 3.7

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

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Step 3.7.1

Simplify the numerator.

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Step 3.7.1.1

Raise $- 6$ to the power of $2$ .

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

Step 3.7.1.2

Multiply $- 4$ by $- 3$ .

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

Step 3.7.1.3

Apply the distributive property.

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

Step 3.7.1.4

Multiply $12$ by $2$ .

$y = \frac{6 \pm \sqrt{36 + 24 + 12 (- x)}}{2 \cdot - 3}$

Step 3.7.1.5

Multiply $- 1$ by $12$ .

$y = \frac{6 \pm \sqrt{36 + 24 - 12 x}}{2 \cdot - 3}$

Step 3.7.1.6

Add $36$ and $24$ .

$y = \frac{6 \pm \sqrt{60 - 12 x}}{2 \cdot - 3}$

Step 3.7.1.7

Factor $12$ out of $60 - 12 x$ .

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Step 3.7.1.7.1

Factor $12$ out of $60$ .

$y = \frac{6 \pm \sqrt{12 (5) - 12 x}}{2 \cdot - 3}$

Step 3.7.1.7.2

Factor $12$ out of $- 12 x$ .

$y = \frac{6 \pm \sqrt{12 (5) + 12 (- x)}}{2 \cdot - 3}$

Step 3.7.1.7.3

Factor $12$ out of $12 (5) + 12 (- x)$ .

$y = \frac{6 \pm \sqrt{12 (5 - x)}}{2 \cdot - 3}$

Step 3.7.1.8

Rewrite $12 (5 - x)$ as $2^{2} \cdot (3 (5 - x))$ .

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Step 3.7.1.8.1

Factor $4$ out of $12$ .

$y = \frac{6 \pm \sqrt{4 (3) (5 - x)}}{2 \cdot - 3}$

Step 3.7.1.8.2

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

$y = \frac{6 \pm \sqrt{2^{2} \cdot (3 (5 - x))}}{2 \cdot - 3}$

Step 3.7.1.8.3

Add parentheses.

$y = \frac{6 \pm \sqrt{2^{2} \cdot (3 (5 - x))}}{2 \cdot - 3}$

Step 3.7.1.9

Pull terms out from under the radical.

$y = \frac{6 \pm 2 \sqrt{3 (5 - x)}}{2 \cdot - 3}$

Step 3.7.2

Multiply $2$ by $- 3$ .

$y = \frac{6 \pm 2 \sqrt{3 (5 - x)}}{- 6}$

Step 3.7.3

Simplify $\frac{6 \pm 2 \sqrt{3 (5 - x)}}{- 6}$ .

$y = \frac{3 \pm \sqrt{3 (5 - x)}}{- 3}$

Step 3.7.4

Move the negative in front of the fraction.

$y = - \frac{3 \pm \sqrt{3 (5 - x)}}{3}$

Step 3.7.5

Change the $\pm$ to $-$ .

$y = - \frac{3 - \sqrt{3 (5 - x)}}{3}$

Step 3.8

The final answer is the combination of both solutions.

$y = - \frac{3 + \sqrt{3 (5 - x)}}{3}$

$y = - \frac{3 - \sqrt{3 (5 - x)}}{3}$

$y = - \frac{3 + \sqrt{3 (5 - x)}}{3}$

$y = - \frac{3 - \sqrt{3 (5 - x)}}{3}$

Step 4

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

$f^{- 1} (x) = - \frac{3 + \sqrt{3 (5 - x)}}{3}, - \frac{3 - \sqrt{3 (5 - x)}}{3}$

Step 5

Verify if $f^{- 1} (x) = - \frac{3 + \sqrt{3 (5 - x)}}{3}, - \frac{3 - \sqrt{3 (5 - x)}}{3}$ is the inverse of $f (x) = - 3 x^{2} - 6 x + 2$ .

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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 $f (x) = - 3 x^{2} - 6 x + 2$ and $f^{- 1} (x) = - \frac{3 + \sqrt{3 (5 - x)}}{3}, - \frac{3 - \sqrt{3 (5 - x)}}{3}$ and compare them.

Step 5.2

Find the range of $f (x) = - 3 x^{2} - 6 x + 2$ .

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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:

$(- \infty, 5]$

Step 5.3

Find the domain of $- \frac{3 + \sqrt{3 (5 - x)}}{3}$ .

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Step 5.3.1

Set the radicand in $\sqrt{3 (5 - x)}$ greater than or equal to $0$ to find where the expression is defined.

$3 (5 - x) \geq 0$

Step 5.3.2

Solve for $x$ .

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Step 5.3.2.1

Divide each term in $3 (5 - x) \geq 0$ by $3$ and simplify.

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Step 5.3.2.1.1

Divide each term in $3 (5 - x) \geq 0$ by $3$ .

$\frac{3 (5 - x)}{3} \geq \frac{0}{3}$

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 $3$ .

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Step 5.3.2.1.2.1.1

Cancel the common factor.

$\frac{3 (5 - x)}{3} \geq \frac{0}{3}$

Step 5.3.2.1.2.1.2

Divide $5 - x$ by $1$ .

$5 - x \geq \frac{0}{3}$

Step 5.3.2.1.3

Simplify the right side.

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Step 5.3.2.1.3.1

Divide $0$ by $3$ .

$5 - x \geq 0$

Step 5.3.2.2

Subtract $5$ from both sides of the inequality.

$- x \geq - 5$

Step 5.3.2.3

Divide each term in $- x \geq - 5$ by $- 1$ and simplify.

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Step 5.3.2.3.1

Divide each term in $- x \geq - 5$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 5}{- 1}$

Step 5.3.2.3.2

Simplify the left side.

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Step 5.3.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 5}{- 1}$

Step 5.3.2.3.2.2

Divide $x$ by $1$ .

$x \leq \frac{- 5}{- 1}$

Step 5.3.2.3.3

Simplify the right side.

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Step 5.3.2.3.3.1

Divide $- 5$ by $- 1$ .

$x \leq 5$

Step 5.3.3

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

$(- \infty, 5]$

Step 5.4

Find the domain of $f (x) = - 3 x^{2} - 6 x + 2$ .

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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 $f^{- 1} (x) = - \frac{3 + \sqrt{3 (5 - x)}}{3}, - \frac{3 - \sqrt{3 (5 - x)}}{3}$ is the range of $f (x) = - 3 x^{2} - 6 x + 2$ and the range of $f^{- 1} (x) = - \frac{3 + \sqrt{3 (5 - x)}}{3}, - \frac{3 - \sqrt{3 (5 - x)}}{3}$ is the domain of $f (x) = - 3 x^{2} - 6 x + 2$ , then $f^{- 1} (x) = - \frac{3 + \sqrt{3 (5 - x)}}{3}, - \frac{3 - \sqrt{3 (5 - x)}}{3}$ is the inverse of $f (x) = - 3 x^{2} - 6 x + 2$ .

$f^{- 1} (x) = - \frac{3 + \sqrt{3 (5 - x)}}{3}, - \frac{3 - \sqrt{3 (5 - x)}}{3}$

Step 6