Finite Math Examples

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

Tap for more steps...

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.

Tap for more steps...

Step 2.5.1

Simplify the numerator.

Tap for more steps...

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 $120 + 8 x$ .

Tap for more steps...

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

Tap for more steps...

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 $\pm$ .

Tap for more steps...

Step 2.6.1

Simplify the numerator.

Tap for more steps...

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 $120 + 8 x$ .

Tap for more steps...

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

Tap for more steps...

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 $\pm$ .

Tap for more steps...

Step 2.7.1

Simplify the numerator.

Tap for more steps...

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 $120 + 8 x$ .

Tap for more steps...

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

Tap for more steps...

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

Tap for more steps...

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

Tap for more steps...

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

Tap for more steps...

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 $x$ .

Tap for more steps...

Step 4.3.2.1

Divide each term in $2 (15 + x) \geq 0$ by $2$ and simplify.

Tap for more steps...

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.

Tap for more steps...

Step 4.3.2.1.2.1

Cancel the common factor of $2$ .

Tap for more steps...

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.

Tap for more steps...

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

Tap for more steps...

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