Finite Math Examples

$f (x) = 3 x^{2} + 18 x + 23$

Step 1

Write $f (x) = 3 x^{2} + 18 x + 23$ as an equation.

$y = 3 x^{2} + 18 x + 23$

Step 2

Interchange the variables.

$x = 3 y^{2} + 18 y + 23$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $3 y^{2} + 18 y + 23 = x$ .

$3 y^{2} + 18 y + 23 = x$

Step 3.2

Subtract $x$ from both sides of the equation.

$3 y^{2} + 18 y + 23 - x = 0$

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 $a = 3$ , $b = 18$ , and $c = 23 - x$ into the quadratic formula and solve for $y$ .

$\frac{- 18 \pm \sqrt{18^{2} - 4 \cdot (3 \cdot (23 - x))}}{2 \cdot 3}$

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 $18$ to the power of $2$ .

$y = \frac{- 18 \pm \sqrt{324 - 4 \cdot 3 \cdot (23 - x)}}{2 \cdot 3}$

Step 3.5.1.2

Multiply $- 4$ by $3$ .

$y = \frac{- 18 \pm \sqrt{324 - 12 \cdot (23 - x)}}{2 \cdot 3}$

Step 3.5.1.3

Apply the distributive property.

$y = \frac{- 18 \pm \sqrt{324 - 12 \cdot 23 - 12 (- x)}}{2 \cdot 3}$

Step 3.5.1.4

Multiply $- 12$ by $23$ .

$y = \frac{- 18 \pm \sqrt{324 - 276 - 12 (- x)}}{2 \cdot 3}$

Step 3.5.1.5

Multiply $- 1$ by $- 12$ .

$y = \frac{- 18 \pm \sqrt{324 - 276 + 12 x}}{2 \cdot 3}$

Step 3.5.1.6

Subtract $276$ from $324$ .

$y = \frac{- 18 \pm \sqrt{48 + 12 x}}{2 \cdot 3}$

Step 3.5.1.7

Factor $12$ out of $48 + 12 x$ .

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Step 3.5.1.7.1

Factor $12$ out of $48$ .

$y = \frac{- 18 \pm \sqrt{12 \cdot 4 + 12 x}}{2 \cdot 3}$

Step 3.5.1.7.2

Factor $12$ out of $12 \cdot 4 + 12 x$ .

$y = \frac{- 18 \pm \sqrt{12 (4 + x)}}{2 \cdot 3}$

Step 3.5.1.8

Rewrite $12 (4 + x)$ as $2^{2} \cdot (3 (2^{2} + x))$ .

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Step 3.5.1.8.1

Factor $4$ out of $12$ .

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

Step 3.5.1.8.2

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

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

Step 3.5.1.8.3

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

$y = \frac{- 18 \pm \sqrt{2^{2} \cdot (3 (2^{2} + x))}}{2 \cdot 3}$

Step 3.5.1.8.4

Add parentheses.

$y = \frac{- 18 \pm \sqrt{2^{2} \cdot (3 (2^{2} + x))}}{2 \cdot 3}$

Step 3.5.1.9

Pull terms out from under the radical.

$y = \frac{- 18 \pm 2 \sqrt{3 (2^{2} + x)}}{2 \cdot 3}$

Step 3.5.1.10

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

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

Step 3.5.2

Multiply $2$ by $3$ .

$y = \frac{- 18 \pm 2 \sqrt{3 (4 + x)}}{6}$

Step 3.5.3

Simplify $\frac{- 18 \pm 2 \sqrt{3 (4 + x)}}{6}$ .

$y = \frac{- 9 \pm \sqrt{3 (4 + x)}}{3}$

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 $18$ to the power of $2$ .

$y = \frac{- 18 \pm \sqrt{324 - 4 \cdot 3 \cdot (23 - x)}}{2 \cdot 3}$

Step 3.6.1.2

Multiply $- 4$ by $3$ .

$y = \frac{- 18 \pm \sqrt{324 - 12 \cdot (23 - x)}}{2 \cdot 3}$

Step 3.6.1.3

Apply the distributive property.

$y = \frac{- 18 \pm \sqrt{324 - 12 \cdot 23 - 12 (- x)}}{2 \cdot 3}$

Step 3.6.1.4

Multiply $- 12$ by $23$ .

$y = \frac{- 18 \pm \sqrt{324 - 276 - 12 (- x)}}{2 \cdot 3}$

Step 3.6.1.5

Multiply $- 1$ by $- 12$ .

$y = \frac{- 18 \pm \sqrt{324 - 276 + 12 x}}{2 \cdot 3}$

Step 3.6.1.6

Subtract $276$ from $324$ .

$y = \frac{- 18 \pm \sqrt{48 + 12 x}}{2 \cdot 3}$

Step 3.6.1.7

Factor $12$ out of $48 + 12 x$ .

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Step 3.6.1.7.1

Factor $12$ out of $48$ .

$y = \frac{- 18 \pm \sqrt{12 \cdot 4 + 12 x}}{2 \cdot 3}$

Step 3.6.1.7.2

Factor $12$ out of $12 \cdot 4 + 12 x$ .

$y = \frac{- 18 \pm \sqrt{12 (4 + x)}}{2 \cdot 3}$

Step 3.6.1.8

Rewrite $12 (4 + x)$ as $2^{2} \cdot (3 (2^{2} + x))$ .

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Step 3.6.1.8.1

Factor $4$ out of $12$ .

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

Step 3.6.1.8.2

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

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

Step 3.6.1.8.3

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

$y = \frac{- 18 \pm \sqrt{2^{2} \cdot (3 (2^{2} + x))}}{2 \cdot 3}$

Step 3.6.1.8.4

Add parentheses.

$y = \frac{- 18 \pm \sqrt{2^{2} \cdot (3 (2^{2} + x))}}{2 \cdot 3}$

Step 3.6.1.9

Pull terms out from under the radical.

$y = \frac{- 18 \pm 2 \sqrt{3 (2^{2} + x)}}{2 \cdot 3}$

Step 3.6.1.10

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

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

Step 3.6.2

Multiply $2$ by $3$ .

$y = \frac{- 18 \pm 2 \sqrt{3 (4 + x)}}{6}$

Step 3.6.3

Simplify $\frac{- 18 \pm 2 \sqrt{3 (4 + x)}}{6}$ .

$y = \frac{- 9 \pm \sqrt{3 (4 + x)}}{3}$

Step 3.6.4

Change the $\pm$ to $+$ .

$y = \frac{- 9 + \sqrt{3 (4 + x)}}{3}$

Step 3.6.5

Rewrite $- 9$ as $- 1 (9)$ .

$y = \frac{- 1 \cdot 9 + \sqrt{3 (4 + x)}}{3}$

Step 3.6.6

Factor $- 1$ out of $\sqrt{3 (4 + x)}$ .

$y = \frac{- 1 \cdot 9 - 1 (- \sqrt{3 (4 + x)})}{3}$

Step 3.6.7

Factor $- 1$ out of $- 1 (9) - 1 (- \sqrt{3 (4 + x)})$ .

$y = \frac{- 1 (9 - \sqrt{3 (4 + x)})}{3}$

Step 3.6.8

Move the negative in front of the fraction.

$y = - \frac{9 - \sqrt{3 (4 + x)}}{3}$

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 $18$ to the power of $2$ .

$y = \frac{- 18 \pm \sqrt{324 - 4 \cdot 3 \cdot (23 - x)}}{2 \cdot 3}$

Step 3.7.1.2

Multiply $- 4$ by $3$ .

$y = \frac{- 18 \pm \sqrt{324 - 12 \cdot (23 - x)}}{2 \cdot 3}$

Step 3.7.1.3

Apply the distributive property.

$y = \frac{- 18 \pm \sqrt{324 - 12 \cdot 23 - 12 (- x)}}{2 \cdot 3}$

Step 3.7.1.4

Multiply $- 12$ by $23$ .

$y = \frac{- 18 \pm \sqrt{324 - 276 - 12 (- x)}}{2 \cdot 3}$

Step 3.7.1.5

Multiply $- 1$ by $- 12$ .

$y = \frac{- 18 \pm \sqrt{324 - 276 + 12 x}}{2 \cdot 3}$

Step 3.7.1.6

Subtract $276$ from $324$ .

$y = \frac{- 18 \pm \sqrt{48 + 12 x}}{2 \cdot 3}$

Step 3.7.1.7

Factor $12$ out of $48 + 12 x$ .

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Step 3.7.1.7.1

Factor $12$ out of $48$ .

$y = \frac{- 18 \pm \sqrt{12 \cdot 4 + 12 x}}{2 \cdot 3}$

Step 3.7.1.7.2

Factor $12$ out of $12 \cdot 4 + 12 x$ .

$y = \frac{- 18 \pm \sqrt{12 (4 + x)}}{2 \cdot 3}$

Step 3.7.1.8

Rewrite $12 (4 + x)$ as $2^{2} \cdot (3 (2^{2} + x))$ .

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Step 3.7.1.8.1

Factor $4$ out of $12$ .

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

Step 3.7.1.8.2

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

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

Step 3.7.1.8.3

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

$y = \frac{- 18 \pm \sqrt{2^{2} \cdot (3 (2^{2} + x))}}{2 \cdot 3}$

Step 3.7.1.8.4

Add parentheses.

$y = \frac{- 18 \pm \sqrt{2^{2} \cdot (3 (2^{2} + x))}}{2 \cdot 3}$

Step 3.7.1.9

Pull terms out from under the radical.

$y = \frac{- 18 \pm 2 \sqrt{3 (2^{2} + x)}}{2 \cdot 3}$

Step 3.7.1.10

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

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

Step 3.7.2

Multiply $2$ by $3$ .

$y = \frac{- 18 \pm 2 \sqrt{3 (4 + x)}}{6}$

Step 3.7.3

Simplify $\frac{- 18 \pm 2 \sqrt{3 (4 + x)}}{6}$ .

$y = \frac{- 9 \pm \sqrt{3 (4 + x)}}{3}$

Step 3.7.4

Change the $\pm$ to $-$ .

$y = \frac{- 9 - \sqrt{3 (4 + x)}}{3}$

Step 3.7.5

Factor $- 1$ out of $- 9 - \sqrt{3 (4 + x)}$ .

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Step 3.7.5.1

Reorder $4$ and $x$ .

$y = \frac{- 9 - \sqrt{3 (x + 4)}}{3}$

Step 3.7.5.2

Rewrite $- 9$ as $- 1 (9)$ .

$y = \frac{- 1 \cdot 9 - \sqrt{3 (x + 4)}}{3}$

Step 3.7.5.3

Factor $- 1$ out of $- \sqrt{3 (x + 4)}$ .

$y = \frac{- 1 \cdot 9 - (\sqrt{3 (x + 4)})}{3}$

Step 3.7.5.4

Factor $- 1$ out of $- 1 (9) - (\sqrt{3 (x + 4)})$ .

$y = \frac{- 1 (9 + \sqrt{3 (x + 4)})}{3}$

Step 3.7.5.5

Rewrite $- 1 (9 + \sqrt{3 (x + 4)})$ as $- (9 + \sqrt{3 (x + 4)})$ .

$y = \frac{- (9 + \sqrt{3 (x + 4)})}{3}$

Step 3.7.6

Move the negative in front of the fraction.

$y = - \frac{9 + \sqrt{3 (x + 4)}}{3}$

Step 3.8

The final answer is the combination of both solutions.

$y = - \frac{9 - \sqrt{3 (4 + x)}}{3}$

$y = - \frac{9 + \sqrt{3 (x + 4)}}{3}$

$y = - \frac{9 - \sqrt{3 (4 + x)}}{3}$

$y = - \frac{9 + \sqrt{3 (x + 4)}}{3}$

Step 4

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

$f^{- 1} (x) = - \frac{9 - \sqrt{3 (4 + x)}}{3}, - \frac{9 + \sqrt{3 (x + 4)}}{3}$

Step 5

Verify if $f^{- 1} (x) = - \frac{9 - \sqrt{3 (4 + x)}}{3}, - \frac{9 + \sqrt{3 (x + 4)}}{3}$ is the inverse of $f (x) = 3 x^{2} + 18 x + 23$ .

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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 $f (x) = 3 x^{2} + 18 x + 23$ and $f^{- 1} (x) = - \frac{9 - \sqrt{3 (4 + x)}}{3}, - \frac{9 + \sqrt{3 (x + 4)}}{3}$ and compare them.

Step 5.2

Find the range of $f (x) = 3 x^{2} + 18 x + 23$ .

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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:

$[- 4, \infty)$

Step 5.3

Find the domain of $- \frac{9 - \sqrt{3 (4 + x)}}{3}$ .

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Step 5.3.1

Set the radicand in $\sqrt{3 (4 + x)}$ greater than or equal to $0$ to find where the expression is defined.

$3 (4 + x) \geq 0$

Step 5.3.2

Solve for $x$ .

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Step 5.3.2.1

Divide each term in $3 (4 + x) \geq 0$ by $3$ and simplify.

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Step 5.3.2.1.1

Divide each term in $3 (4 + x) \geq 0$ by $3$ .

$\frac{3 (4 + x)}{3} \geq \frac{0}{3}$

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.

$\frac{3 (4 + x)}{3} \geq \frac{0}{3}$

Step 5.3.2.1.2.1.2

Divide $4 + x$ by $1$ .

$4 + x \geq \frac{0}{3}$

Step 5.3.2.1.3

Simplify the right side.

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Step 5.3.2.1.3.1

Divide $0$ by $3$ .

$4 + x \geq 0$

Step 5.3.2.2

Subtract $4$ from both sides of the inequality.

$x \geq - 4$

Step 5.3.3

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

$[- 4, \infty)$

Step 5.4

Find the domain of $f (x) = 3 x^{2} + 18 x + 23$ .

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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 $f^{- 1} (x) = - \frac{9 - \sqrt{3 (4 + x)}}{3}, - \frac{9 + \sqrt{3 (x + 4)}}{3}$ is the range of $f (x) = 3 x^{2} + 18 x + 23$ and the range of $f^{- 1} (x) = - \frac{9 - \sqrt{3 (4 + x)}}{3}, - \frac{9 + \sqrt{3 (x + 4)}}{3}$ is the domain of $f (x) = 3 x^{2} + 18 x + 23$ , then $f^{- 1} (x) = - \frac{9 - \sqrt{3 (4 + x)}}{3}, - \frac{9 + \sqrt{3 (x + 4)}}{3}$ is the inverse of $f (x) = 3 x^{2} + 18 x + 23$ .

$f^{- 1} (x) = - \frac{9 - \sqrt{3 (4 + x)}}{3}, - \frac{9 + \sqrt{3 (x + 4)}}{3}$

Step 6