Finite Math Examples

2 x^{2} - 12 x + 3

$2 x^{2} - 12 x + 3$

Step 1

Interchange the variables.

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

Step 2

Solve for

$y$ .

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Step 2.1

Rewrite the equation as

$2 y^{2} - 12 y + 3 = x$ .

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

Step 2.2

Subtract

$x$ from both sides of the equation.

$2 y^{2} - 12 y + 3 - x = 0$

Step 2.3

Use the quadratic formula to find the solutions.

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

Step 2.4

Substitute the values

$a = 2$ ,

$b = - 12$ , and

$c = 3 - x$ into the quadratic formula and solve for

$y$ .

$\frac{12 \pm \sqrt{{(- 12)}^{2} - 4 \cdot (2 \cdot (3 - x))}}{2 \cdot 2}$

Step 2.5

Simplify.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise

$- 12$ to the power of

$2$ .

$y = \frac{12 \pm \sqrt{144 - 4 \cdot 2 \cdot (3 - x)}}{2 \cdot 2}$

Step 2.5.1.2

Multiply

$- 4$ by

$2$ .

$y = \frac{12 \pm \sqrt{144 - 8 \cdot (3 - x)}}{2 \cdot 2}$

Step 2.5.1.3

Apply the distributive property.

$y = \frac{12 \pm \sqrt{144 - 8 \cdot 3 - 8 (- x)}}{2 \cdot 2}$

Step 2.5.1.4

Multiply

$- 8$ by

$3$ .

$y = \frac{12 \pm \sqrt{144 - 24 - 8 (- x)}}{2 \cdot 2}$

Step 2.5.1.5

Multiply

$- 1$ by

$- 8$ .

$y = \frac{12 \pm \sqrt{144 - 24 + 8 x}}{2 \cdot 2}$

Step 2.5.1.6

Subtract

$24$ from

$144$ .

$y = \frac{12 \pm \sqrt{120 + 8 x}}{2 \cdot 2}$

Step 2.5.1.7

Factor

$8$ out of

$120 + 8 x$ .

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Step 2.5.1.7.1

Factor

$8$ out of

$120$ .

$y = \frac{12 \pm \sqrt{8 \cdot 15 + 8 x}}{2 \cdot 2}$

Step 2.5.1.7.2

Factor

$8$ out of

$8 \cdot 15 + 8 x$ .

$y = \frac{12 \pm \sqrt{8 (15 + x)}}{2 \cdot 2}$

Step 2.5.1.8

Rewrite

$8 (15 + x)$ as

$2^{2} \cdot (2 (15 + x))$ .

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Step 2.5.1.8.1

Factor

$4$ out of

$8$ .

$y = \frac{12 \pm \sqrt{4 (2) (15 + x)}}{2 \cdot 2}$

Step 2.5.1.8.2

Rewrite

$4$ as

$2^{2}$ .

$y = \frac{12 \pm \sqrt{2^{2} \cdot (2 (15 + x))}}{2 \cdot 2}$

Step 2.5.1.8.3

Add parentheses.

$y = \frac{12 \pm \sqrt{2^{2} \cdot (2 (15 + x))}}{2 \cdot 2}$

Step 2.5.1.9

Pull terms out from under the radical.

$y = \frac{12 \pm 2 \sqrt{2 (15 + x)}}{2 \cdot 2}$

Step 2.5.2

Multiply

$2$ by

$2$ .

$y = \frac{12 \pm 2 \sqrt{2 (15 + x)}}{4}$

Step 2.5.3

Simplify

$\frac{12 \pm 2 \sqrt{2 (15 + x)}}{4}$ .

$y = \frac{6 \pm \sqrt{2 (15 + x)}}{2}$

Step 2.6

Simplify the expression to solve for the

$+$ portion of the

$\pm$ .

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Raise

$- 12$ to the power of

$2$ .

$y = \frac{12 \pm \sqrt{144 - 4 \cdot 2 \cdot (3 - x)}}{2 \cdot 2}$

Step 2.6.1.2

Multiply

$- 4$ by

$2$ .

$y = \frac{12 \pm \sqrt{144 - 8 \cdot (3 - x)}}{2 \cdot 2}$

Step 2.6.1.3

Apply the distributive property.

$y = \frac{12 \pm \sqrt{144 - 8 \cdot 3 - 8 (- x)}}{2 \cdot 2}$

Step 2.6.1.4

Multiply

$- 8$ by

$3$ .

$y = \frac{12 \pm \sqrt{144 - 24 - 8 (- x)}}{2 \cdot 2}$

Step 2.6.1.5

Multiply

$- 1$ by

$- 8$ .

$y = \frac{12 \pm \sqrt{144 - 24 + 8 x}}{2 \cdot 2}$

Step 2.6.1.6

Subtract

$24$ from

$144$ .

$y = \frac{12 \pm \sqrt{120 + 8 x}}{2 \cdot 2}$

Step 2.6.1.7

Factor

$8$ out of

$120 + 8 x$ .

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Step 2.6.1.7.1

Factor

$8$ out of

$120$ .

$y = \frac{12 \pm \sqrt{8 \cdot 15 + 8 x}}{2 \cdot 2}$

Step 2.6.1.7.2

Factor

$8$ out of

$8 \cdot 15 + 8 x$ .

$y = \frac{12 \pm \sqrt{8 (15 + x)}}{2 \cdot 2}$

Step 2.6.1.8

Rewrite

$8 (15 + x)$ as

$2^{2} \cdot (2 (15 + x))$ .

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Step 2.6.1.8.1

Factor

$4$ out of

$8$ .

$y = \frac{12 \pm \sqrt{4 (2) (15 + x)}}{2 \cdot 2}$

Step 2.6.1.8.2

Rewrite

$4$ as

$2^{2}$ .

$y = \frac{12 \pm \sqrt{2^{2} \cdot (2 (15 + x))}}{2 \cdot 2}$

Step 2.6.1.8.3

Add parentheses.

$y = \frac{12 \pm \sqrt{2^{2} \cdot (2 (15 + x))}}{2 \cdot 2}$

Step 2.6.1.9

Pull terms out from under the radical.

$y = \frac{12 \pm 2 \sqrt{2 (15 + x)}}{2 \cdot 2}$

Step 2.6.2

Multiply

$2$ by

$2$ .

$y = \frac{12 \pm 2 \sqrt{2 (15 + x)}}{4}$

Step 2.6.3

Simplify

$\frac{12 \pm 2 \sqrt{2 (15 + x)}}{4}$ .

$y = \frac{6 \pm \sqrt{2 (15 + x)}}{2}$

Step 2.6.4

Change the

$\pm$ to

$+$ .

$y = \frac{6 + \sqrt{2 (15 + x)}}{2}$

Step 2.7

Simplify the expression to solve for the

$-$ portion of the

$\pm$ .

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

Raise

$- 12$ to the power of

$2$ .

$y = \frac{12 \pm \sqrt{144 - 4 \cdot 2 \cdot (3 - x)}}{2 \cdot 2}$

Step 2.7.1.2

Multiply

$- 4$ by

$2$ .

$y = \frac{12 \pm \sqrt{144 - 8 \cdot (3 - x)}}{2 \cdot 2}$

Step 2.7.1.3

Apply the distributive property.

$y = \frac{12 \pm \sqrt{144 - 8 \cdot 3 - 8 (- x)}}{2 \cdot 2}$

Step 2.7.1.4

Multiply

$- 8$ by

$3$ .

$y = \frac{12 \pm \sqrt{144 - 24 - 8 (- x)}}{2 \cdot 2}$

Step 2.7.1.5

Multiply

$- 1$ by

$- 8$ .

$y = \frac{12 \pm \sqrt{144 - 24 + 8 x}}{2 \cdot 2}$

Step 2.7.1.6

Subtract

$24$ from

$144$ .

$y = \frac{12 \pm \sqrt{120 + 8 x}}{2 \cdot 2}$

Step 2.7.1.7

Factor

$8$ out of

$120 + 8 x$ .

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Step 2.7.1.7.1

Factor

$8$ out of

$120$ .

$y = \frac{12 \pm \sqrt{8 \cdot 15 + 8 x}}{2 \cdot 2}$

Step 2.7.1.7.2

Factor

$8$ out of

$8 \cdot 15 + 8 x$ .

$y = \frac{12 \pm \sqrt{8 (15 + x)}}{2 \cdot 2}$

Step 2.7.1.8

Rewrite

$8 (15 + x)$ as

$2^{2} \cdot (2 (15 + x))$ .

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Step 2.7.1.8.1

Factor

$4$ out of

$8$ .

$y = \frac{12 \pm \sqrt{4 (2) (15 + x)}}{2 \cdot 2}$

Step 2.7.1.8.2

Rewrite

$4$ as

$2^{2}$ .

$y = \frac{12 \pm \sqrt{2^{2} \cdot (2 (15 + x))}}{2 \cdot 2}$

Step 2.7.1.8.3

Add parentheses.

$y = \frac{12 \pm \sqrt{2^{2} \cdot (2 (15 + x))}}{2 \cdot 2}$

Step 2.7.1.9

Pull terms out from under the radical.

$y = \frac{12 \pm 2 \sqrt{2 (15 + x)}}{2 \cdot 2}$

Step 2.7.2

Multiply

$2$ by

$2$ .

$y = \frac{12 \pm 2 \sqrt{2 (15 + x)}}{4}$

Step 2.7.3

Simplify

$\frac{12 \pm 2 \sqrt{2 (15 + x)}}{4}$ .

$y = \frac{6 \pm \sqrt{2 (15 + x)}}{2}$

Step 2.7.4

Change the

$\pm$ to

$-$ .

$y = \frac{6 - \sqrt{2 (15 + x)}}{2}$

Step 2.8

The final answer is the combination of both solutions.

$y = \frac{6 + \sqrt{2 (15 + x)}}{2}$

$y = \frac{6 - \sqrt{2 (15 + x)}}{2}$

$y = \frac{6 + \sqrt{2 (15 + x)}}{2}$

$y = \frac{6 - \sqrt{2 (15 + x)}}{2}$

Step 3

Replace

$y$ with

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

$f^{- 1} (x) = \frac{6 + \sqrt{2 (15 + x)}}{2}, \frac{6 - \sqrt{2 (15 + x)}}{2}$

Step 4

Verify if

$f^{- 1} (x) = \frac{6 + \sqrt{2 (15 + x)}}{2}, \frac{6 - \sqrt{2 (15 + x)}}{2}$ is the inverse of

$f (x) = 2 x^{2} - 12 x + 3$ .

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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

$f (x) = 2 x^{2} - 12 x + 3$ and

$f^{- 1} (x) = \frac{6 + \sqrt{2 (15 + x)}}{2}, \frac{6 - \sqrt{2 (15 + x)}}{2}$ and compare them.

Step 4.2

Find the range of

$f (x) = 2 x^{2} - 12 x + 3$ .

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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:

$[- 15, \infty)$

Step 4.3

Find the domain of

$\frac{6 + \sqrt{2 (15 + x)}}{2}$ .

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Step 4.3.1

Set the radicand in

$\sqrt{2 (15 + x)}$ greater than or equal to

$0$ to find where the expression is defined.

$2 (15 + x) \geq 0$

Step 4.3.2

Solve for

$x$ .

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Step 4.3.2.1

Divide each term in

$2 (15 + x) \geq 0$ by

$2$ and simplify.

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Step 4.3.2.1.1

Divide each term in

$2 (15 + x) \geq 0$ by

$2$ .

$\frac{2 (15 + x)}{2} \geq \frac{0}{2}$

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

$2$ .

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Step 4.3.2.1.2.1.1

Cancel the common factor.

$\frac{2 (15 + x)}{2} \geq \frac{0}{2}$

Step 4.3.2.1.2.1.2

Divide

$15 + x$ by

$1$ .

$15 + x \geq \frac{0}{2}$

Step 4.3.2.1.3

Simplify the right side.

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Step 4.3.2.1.3.1

Divide

$0$ by

$2$ .

$15 + x \geq 0$

Step 4.3.2.2

Subtract

$15$ from both sides of the inequality.

$x \geq - 15$

Step 4.3.3

The domain is all values of

$x$ that make the expression defined.

$[- 15, \infty)$

Step 4.4

Find the domain of

$f (x) = 2 x^{2} - 12 x + 3$ .

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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

$f^{- 1} (x) = \frac{6 + \sqrt{2 (15 + x)}}{2}, \frac{6 - \sqrt{2 (15 + x)}}{2}$ is the range of

$f (x) = 2 x^{2} - 12 x + 3$ and the range of

$f^{- 1} (x) = \frac{6 + \sqrt{2 (15 + x)}}{2}, \frac{6 - \sqrt{2 (15 + x)}}{2}$ is the domain of

$f (x) = 2 x^{2} - 12 x + 3$ , then

$f^{- 1} (x) = \frac{6 + \sqrt{2 (15 + x)}}{2}, \frac{6 - \sqrt{2 (15 + x)}}{2}$ is the inverse of

$f (x) = 2 x^{2} - 12 x + 3$ .

$f^{- 1} (x) = \frac{6 + \sqrt{2 (15 + x)}}{2}, \frac{6 - \sqrt{2 (15 + x)}}{2}$

Step 5