Finite Math Examples

$f (x) = \frac{x - 9}{(x - 7) (x + 1)}$

Step 1

Write $f (x) = \frac{x - 9}{(x - 7) (x + 1)}$ as an equation.

$y = \frac{x - 9}{(x - 7) (x + 1)}$

Step 2

Interchange the variables.

$x = \frac{y - 9}{(y - 7) (y + 1)}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{y - 9}{(y - 7) (y + 1)} = x$ .

$\frac{y - 9}{(y - 7) (y + 1)} = x$

Step 3.2

Find the LCD of the terms in the equation.

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Step 3.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$(y - 7) (y + 1), 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$(y - 7) (y + 1)$

Step 3.3

Multiply each term in $\frac{y - 9}{(y - 7) (y + 1)} = x$ by $(y - 7) (y + 1)$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{y - 9}{(y - 7) (y + 1)} = x$ by $(y - 7) (y + 1)$ .

$\frac{y - 9}{(y - 7) (y + 1)} ((y - 7) (y + 1)) = x ((y - 7) (y + 1))$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $(y - 7) (y + 1)$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{y - 9}{(y - 7) (y + 1)} ((y - 7) (y + 1)) = x ((y - 7) (y + 1))$

Step 3.3.2.1.2

Rewrite the expression.

$y - 9 = x ((y - 7) (y + 1))$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Expand $(y - 7) (y + 1)$ using the FOIL Method.

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Step 3.3.3.1.1

Apply the distributive property.

$y - 9 = x (y (y + 1) - 7 (y + 1))$

Step 3.3.3.1.2

Apply the distributive property.

$y - 9 = x (y \cdot y + y \cdot 1 - 7 (y + 1))$

Step 3.3.3.1.3

Apply the distributive property.

$y - 9 = x (y \cdot y + y \cdot 1 - 7 y - 7 \cdot 1)$

Step 3.3.3.2

Simplify and combine like terms.

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Step 3.3.3.2.1

Simplify each term.

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Step 3.3.3.2.1.1

Multiply $y$ by $y$ .

$y - 9 = x (y^{2} + y \cdot 1 - 7 y - 7 \cdot 1)$

Step 3.3.3.2.1.2

Multiply $y$ by $1$ .

$y - 9 = x (y^{2} + y - 7 y - 7 \cdot 1)$

Step 3.3.3.2.1.3

Multiply $- 7$ by $1$ .

$y - 9 = x (y^{2} + y - 7 y - 7)$

Step 3.3.3.2.2

Subtract $7 y$ from $y$ .

$y - 9 = x (y^{2} - 6 y - 7)$

Step 3.3.3.3

Apply the distributive property.

$y - 9 = x y^{2} + x (- 6 y) + x \cdot - 7$

Step 3.3.3.4

Simplify.

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Step 3.3.3.4.1

Rewrite using the commutative property of multiplication.

$y - 9 = x y^{2} - 6 x y + x \cdot - 7$

Step 3.3.3.4.2

Move $- 7$ to the left of $x$ .

$y - 9 = x y^{2} - 6 x y - 7 \cdot x$

$y - 9 = x y^{2} - 6 x y - 7 x$

Step 3.4

Solve the equation.

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Step 3.4.1

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x y^{2} - 6 x y - 7 x = y - 9$

Step 3.4.2

Subtract $y$ from both sides of the equation.

$x y^{2} - 6 x y - 7 x - y = - 9$

Step 3.4.3

Add $9$ to both sides of the equation.

$x y^{2} - 6 x y - 7 x - y + 9 = 0$

Step 3.4.4

Use the quadratic formula to find the solutions.

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

Step 3.4.5

Substitute the values $a = x$ , $b = - 6 x - 1$ , and $c = - 7 x + 9$ into the quadratic formula and solve for $y$ .

$\frac{- (- 6 x - 1) \pm \sqrt{{(- 6 x - 1)}^{2} - 4 \cdot (x \cdot (- 7 x + 9))}}{2 x}$

Step 3.4.6

Simplify the numerator.

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Step 3.4.6.1

Apply the distributive property.

$y = \frac{- (- 6 x) + 1 \pm \sqrt{{(- 6 x - 1)}^{2} - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.6.2

Multiply $- 6$ by $- 1$ .

$y = \frac{6 x + 1 \pm \sqrt{{(- 6 x - 1)}^{2} - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.6.3

Multiply $- 1$ by $- 1$ .

$y = \frac{6 x + 1 \pm \sqrt{{(- 6 x - 1)}^{2} - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.6.4

Rewrite ${(- 6 x - 1)}^{2}$ as $(- 6 x - 1) (- 6 x - 1)$ .

$y = \frac{6 x + 1 \pm \sqrt{(- 6 x - 1) (- 6 x - 1) - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.6.5

Expand $(- 6 x - 1) (- 6 x - 1)$ using the FOIL Method.

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Step 3.4.6.5.1

Apply the distributive property.

$y = \frac{6 x + 1 \pm \sqrt{- 6 x (- 6 x - 1) - 1 (- 6 x - 1) - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.6.5.2

Apply the distributive property.

$y = \frac{6 x + 1 \pm \sqrt{- 6 x (- 6 x) - 6 x \cdot - 1 - 1 (- 6 x - 1) - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.6.5.3

Apply the distributive property.

$y = \frac{6 x + 1 \pm \sqrt{- 6 x (- 6 x) - 6 x \cdot - 1 - 1 (- 6 x) - 1 \cdot - 1 - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.6.6

Simplify and combine like terms.

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Step 3.4.6.6.1

Simplify each term.

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Step 3.4.6.6.1.1

Rewrite using the commutative property of multiplication.

$y = \frac{6 x + 1 \pm \sqrt{- 6 \cdot (- 6 x \cdot x) - 6 x \cdot - 1 - 1 (- 6 x) - 1 \cdot - 1 - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.6.6.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.4.6.6.1.2.1

Move $x$ .

$y = \frac{6 x + 1 \pm \sqrt{- 6 \cdot (- 6 (x \cdot x)) - 6 x \cdot - 1 - 1 (- 6 x) - 1 \cdot - 1 - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.6.6.1.2.2

Multiply $x$ by $x$ .

$y = \frac{6 x + 1 \pm \sqrt{- 6 \cdot (- 6 x^{2}) - 6 x \cdot - 1 - 1 (- 6 x) - 1 \cdot - 1 - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.6.6.1.3

Multiply $- 6$ by $- 6$ .

$y = \frac{6 x + 1 \pm \sqrt{36 x^{2} - 6 x \cdot - 1 - 1 (- 6 x) - 1 \cdot - 1 - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.6.6.1.4

Multiply $- 1$ by $- 6$ .

$y = \frac{6 x + 1 \pm \sqrt{36 x^{2} + 6 x - 1 (- 6 x) - 1 \cdot - 1 - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.6.6.1.5

Multiply $- 6$ by $- 1$ .

$y = \frac{6 x + 1 \pm \sqrt{36 x^{2} + 6 x + 6 x - 1 \cdot - 1 - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.6.6.1.6

Multiply $- 1$ by $- 1$ .

$y = \frac{6 x + 1 \pm \sqrt{36 x^{2} + 6 x + 6 x + 1 - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.6.6.2

Add $6 x$ and $6 x$ .

$y = \frac{6 x + 1 \pm \sqrt{36 x^{2} + 12 x + 1 - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.6.7

Apply the distributive property.

$y = \frac{6 x + 1 \pm \sqrt{36 x^{2} + 12 x + 1 - 4 x (- 7 x) - 4 x \cdot 9}}{2 x}$

Step 3.4.6.8

Rewrite using the commutative property of multiplication.

$y = \frac{6 x + 1 \pm \sqrt{36 x^{2} + 12 x + 1 - 4 \cdot (- 7 x \cdot x) - 4 x \cdot 9}}{2 x}$

Step 3.4.6.9

Multiply $9$ by $- 4$ .

$y = \frac{6 x + 1 \pm \sqrt{36 x^{2} + 12 x + 1 - 4 \cdot (- 7 x \cdot x) - 36 x}}{2 x}$

Step 3.4.6.10

Simplify each term.

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Step 3.4.6.10.1

Multiply $x$ by $x$ by adding the exponents.

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Step 3.4.6.10.1.1

Move $x$ .

$y = \frac{6 x + 1 \pm \sqrt{36 x^{2} + 12 x + 1 - 4 \cdot (- 7 (x \cdot x)) - 36 x}}{2 x}$

Step 3.4.6.10.1.2

Multiply $x$ by $x$ .

$y = \frac{6 x + 1 \pm \sqrt{36 x^{2} + 12 x + 1 - 4 \cdot (- 7 x^{2}) - 36 x}}{2 x}$

Step 3.4.6.10.2

Multiply $- 4$ by $- 7$ .

$y = \frac{6 x + 1 \pm \sqrt{36 x^{2} + 12 x + 1 + 28 x^{2} - 36 x}}{2 x}$

Step 3.4.6.11

Add $36 x^{2}$ and $28 x^{2}$ .

$y = \frac{6 x + 1 \pm \sqrt{64 x^{2} + 12 x + 1 - 36 x}}{2 x}$

Step 3.4.6.12

Subtract $36 x$ from $12 x$ .

$y = \frac{6 x + 1 \pm \sqrt{64 x^{2} - 24 x + 1}}{2 x}$

Step 3.4.7

Change the $\pm$ to $+$ .

$y = \frac{6 x + 1 + \sqrt{64 x^{2} - 24 x + 1}}{2 x}$

Step 3.4.8

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

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Step 3.4.8.1

Simplify the numerator.

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Step 3.4.8.1.1

Apply the distributive property.

$y = \frac{- (- 6 x) + 1 \pm \sqrt{{(- 6 x - 1)}^{2} - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.8.1.2

Multiply $- 6$ by $- 1$ .

$y = \frac{6 x + 1 \pm \sqrt{{(- 6 x - 1)}^{2} - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.8.1.3

Multiply $- 1$ by $- 1$ .

$y = \frac{6 x + 1 \pm \sqrt{{(- 6 x - 1)}^{2} - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.8.1.4

Rewrite ${(- 6 x - 1)}^{2}$ as $(- 6 x - 1) (- 6 x - 1)$ .

$y = \frac{6 x + 1 \pm \sqrt{(- 6 x - 1) (- 6 x - 1) - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.8.1.5

Expand $(- 6 x - 1) (- 6 x - 1)$ using the FOIL Method.

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Step 3.4.8.1.5.1

Apply the distributive property.

$y = \frac{6 x + 1 \pm \sqrt{- 6 x (- 6 x - 1) - 1 (- 6 x - 1) - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.8.1.5.2

Apply the distributive property.

$y = \frac{6 x + 1 \pm \sqrt{- 6 x (- 6 x) - 6 x \cdot - 1 - 1 (- 6 x - 1) - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.8.1.5.3

Apply the distributive property.

$y = \frac{6 x + 1 \pm \sqrt{- 6 x (- 6 x) - 6 x \cdot - 1 - 1 (- 6 x) - 1 \cdot - 1 - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.8.1.6

Simplify and combine like terms.

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Step 3.4.8.1.6.1

Simplify each term.

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Step 3.4.8.1.6.1.1

Rewrite using the commutative property of multiplication.

$y = \frac{6 x + 1 \pm \sqrt{- 6 \cdot (- 6 x \cdot x) - 6 x \cdot - 1 - 1 (- 6 x) - 1 \cdot - 1 - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.8.1.6.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.4.8.1.6.1.2.1

Move $x$ .

$y = \frac{6 x + 1 \pm \sqrt{- 6 \cdot (- 6 (x \cdot x)) - 6 x \cdot - 1 - 1 (- 6 x) - 1 \cdot - 1 - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.8.1.6.1.2.2

Multiply $x$ by $x$ .

$y = \frac{6 x + 1 \pm \sqrt{- 6 \cdot (- 6 x^{2}) - 6 x \cdot - 1 - 1 (- 6 x) - 1 \cdot - 1 - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.8.1.6.1.3

Multiply $- 6$ by $- 6$ .

$y = \frac{6 x + 1 \pm \sqrt{36 x^{2} - 6 x \cdot - 1 - 1 (- 6 x) - 1 \cdot - 1 - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.8.1.6.1.4

Multiply $- 1$ by $- 6$ .

$y = \frac{6 x + 1 \pm \sqrt{36 x^{2} + 6 x - 1 (- 6 x) - 1 \cdot - 1 - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.8.1.6.1.5

Multiply $- 6$ by $- 1$ .

$y = \frac{6 x + 1 \pm \sqrt{36 x^{2} + 6 x + 6 x - 1 \cdot - 1 - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.8.1.6.1.6

Multiply $- 1$ by $- 1$ .

$y = \frac{6 x + 1 \pm \sqrt{36 x^{2} + 6 x + 6 x + 1 - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.8.1.6.2

Add $6 x$ and $6 x$ .

$y = \frac{6 x + 1 \pm \sqrt{36 x^{2} + 12 x + 1 - 4 \cdot x \cdot (- 7 x + 9)}}{2 x}$

Step 3.4.8.1.7

Apply the distributive property.

$y = \frac{6 x + 1 \pm \sqrt{36 x^{2} + 12 x + 1 - 4 x (- 7 x) - 4 x \cdot 9}}{2 x}$

Step 3.4.8.1.8

Rewrite using the commutative property of multiplication.

$y = \frac{6 x + 1 \pm \sqrt{36 x^{2} + 12 x + 1 - 4 \cdot (- 7 x \cdot x) - 4 x \cdot 9}}{2 x}$

Step 3.4.8.1.9

Multiply $9$ by $- 4$ .

$y = \frac{6 x + 1 \pm \sqrt{36 x^{2} + 12 x + 1 - 4 \cdot (- 7 x \cdot x) - 36 x}}{2 x}$

Step 3.4.8.1.10

Simplify each term.

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Step 3.4.8.1.10.1

Multiply $x$ by $x$ by adding the exponents.

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Step 3.4.8.1.10.1.1

Move $x$ .

$y = \frac{6 x + 1 \pm \sqrt{36 x^{2} + 12 x + 1 - 4 \cdot (- 7 (x \cdot x)) - 36 x}}{2 x}$

Step 3.4.8.1.10.1.2

Multiply $x$ by $x$ .

$y = \frac{6 x + 1 \pm \sqrt{36 x^{2} + 12 x + 1 - 4 \cdot (- 7 x^{2}) - 36 x}}{2 x}$

Step 3.4.8.1.10.2

Multiply $- 4$ by $- 7$ .

$y = \frac{6 x + 1 \pm \sqrt{36 x^{2} + 12 x + 1 + 28 x^{2} - 36 x}}{2 x}$

Step 3.4.8.1.11

Add $36 x^{2}$ and $28 x^{2}$ .

$y = \frac{6 x + 1 \pm \sqrt{64 x^{2} + 12 x + 1 - 36 x}}{2 x}$

Step 3.4.8.1.12

Subtract $36 x$ from $12 x$ .

$y = \frac{6 x + 1 \pm \sqrt{64 x^{2} - 24 x + 1}}{2 x}$

Step 3.4.8.2

Change the $\pm$ to $-$ .

$y = \frac{6 x + 1 - \sqrt{64 x^{2} - 24 x + 1}}{2 x}$

Step 3.4.9

The final answer is the combination of both solutions.

$y = \frac{6 x + 1 + \sqrt{64 x^{2} - 24 x + 1}}{2 x}$

$y = \frac{6 x + 1 - \sqrt{64 x^{2} - 24 x + 1}}{2 x}$

$y = \frac{6 x + 1 + \sqrt{64 x^{2} - 24 x + 1}}{2 x}$

$y = \frac{6 x + 1 - \sqrt{64 x^{2} - 24 x + 1}}{2 x}$

$y = \frac{6 x + 1 + \sqrt{64 x^{2} - 24 x + 1}}{2 x}$

$y = \frac{6 x + 1 - \sqrt{64 x^{2} - 24 x + 1}}{2 x}$

Step 4

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

$f^{- 1} (x) = \frac{6 x + 1 + \sqrt{64 x^{2} - 24 x + 1}}{2 x}, \frac{6 x + 1 - \sqrt{64 x^{2} - 24 x + 1}}{2 x}$

Step 5

Verify if $f^{- 1} (x) = \frac{6 x + 1 + \sqrt{64 x^{2} - 24 x + 1}}{2 x}, \frac{6 x + 1 - \sqrt{64 x^{2} - 24 x + 1}}{2 x}$ is the inverse of $f (x) = \frac{x - 9}{(x - 7) (x + 1)}$ .

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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 $f (x) = \frac{x - 9}{(x - 7) (x + 1)}$ and $f^{- 1} (x) = \frac{6 x + 1 + \sqrt{64 x^{2} - 24 x + 1}}{2 x}, \frac{6 x + 1 - \sqrt{64 x^{2} - 24 x + 1}}{2 x}$ and compare them.

Step 5.2

Find the range of $f (x) = \frac{x - 9}{(x - 7) (x + 1)}$ .

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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:

$(- \infty, \frac{3 - \sqrt{5}}{16}] \cup [\frac{3 + \sqrt{5}}{16}, \infty)$

Step 5.3

Find the domain of $\frac{6 x + 1 + \sqrt{64 x^{2} - 24 x + 1}}{2 x}$ .

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Step 5.3.1

Set the radicand in $\sqrt{64 x^{2} - 24 x + 1}$ greater than or equal to $0$ to find where the expression is defined.

$64 x^{2} - 24 x + 1 \geq 0$

Step 5.3.2

Solve for $x$ .

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Step 5.3.2.1

Convert the inequality to an equation.

$64 x^{2} - 24 x + 1 = 0$

Step 5.3.2.2

Use the quadratic formula to find the solutions.

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

Step 5.3.2.3

Substitute the values $a = 64$ , $b = - 24$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{24 \pm \sqrt{{(- 24)}^{2} - 4 \cdot (64 \cdot 1)}}{2 \cdot 64}$

Step 5.3.2.4

Simplify.

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Step 5.3.2.4.1

Simplify the numerator.

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Step 5.3.2.4.1.1

Raise $- 24$ to the power of $2$ .

$x = \frac{24 \pm \sqrt{576 - 4 \cdot 64 \cdot 1}}{2 \cdot 64}$

Step 5.3.2.4.1.2

Multiply $- 4 \cdot 64 \cdot 1$ .

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Step 5.3.2.4.1.2.1

Multiply $- 4$ by $64$ .

$x = \frac{24 \pm \sqrt{576 - 256 \cdot 1}}{2 \cdot 64}$

Step 5.3.2.4.1.2.2

Multiply $- 256$ by $1$ .

$x = \frac{24 \pm \sqrt{576 - 256}}{2 \cdot 64}$

Step 5.3.2.4.1.3

Subtract $256$ from $576$ .

$x = \frac{24 \pm \sqrt{320}}{2 \cdot 64}$

Step 5.3.2.4.1.4

Rewrite $320$ as $8^{2} \cdot 5$ .

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Step 5.3.2.4.1.4.1

Factor $64$ out of $320$ .

$x = \frac{24 \pm \sqrt{64 (5)}}{2 \cdot 64}$

Step 5.3.2.4.1.4.2

Rewrite $64$ as $8^{2}$ .

$x = \frac{24 \pm \sqrt{8^{2} \cdot 5}}{2 \cdot 64}$

Step 5.3.2.4.1.5

Pull terms out from under the radical.

$x = \frac{24 \pm 8 \sqrt{5}}{2 \cdot 64}$

Step 5.3.2.4.2

Multiply $2$ by $64$ .

$x = \frac{24 \pm 8 \sqrt{5}}{128}$

Step 5.3.2.4.3

Simplify $\frac{24 \pm 8 \sqrt{5}}{128}$ .

$x = \frac{3 \pm \sqrt{5}}{16}$

Step 5.3.2.5

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

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Step 5.3.2.5.1

Simplify the numerator.

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Step 5.3.2.5.1.1

Raise $- 24$ to the power of $2$ .

$x = \frac{24 \pm \sqrt{576 - 4 \cdot 64 \cdot 1}}{2 \cdot 64}$

Step 5.3.2.5.1.2

Multiply $- 4 \cdot 64 \cdot 1$ .

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Step 5.3.2.5.1.2.1

Multiply $- 4$ by $64$ .

$x = \frac{24 \pm \sqrt{576 - 256 \cdot 1}}{2 \cdot 64}$

Step 5.3.2.5.1.2.2

Multiply $- 256$ by $1$ .

$x = \frac{24 \pm \sqrt{576 - 256}}{2 \cdot 64}$

Step 5.3.2.5.1.3

Subtract $256$ from $576$ .

$x = \frac{24 \pm \sqrt{320}}{2 \cdot 64}$

Step 5.3.2.5.1.4

Rewrite $320$ as $8^{2} \cdot 5$ .

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Step 5.3.2.5.1.4.1

Factor $64$ out of $320$ .

$x = \frac{24 \pm \sqrt{64 (5)}}{2 \cdot 64}$

Step 5.3.2.5.1.4.2

Rewrite $64$ as $8^{2}$ .

$x = \frac{24 \pm \sqrt{8^{2} \cdot 5}}{2 \cdot 64}$

Step 5.3.2.5.1.5

Pull terms out from under the radical.

$x = \frac{24 \pm 8 \sqrt{5}}{2 \cdot 64}$

Step 5.3.2.5.2

Multiply $2$ by $64$ .

$x = \frac{24 \pm 8 \sqrt{5}}{128}$

Step 5.3.2.5.3

Simplify $\frac{24 \pm 8 \sqrt{5}}{128}$ .

$x = \frac{3 \pm \sqrt{5}}{16}$

Step 5.3.2.5.4

Change the $\pm$ to $+$ .

$x = \frac{3 + \sqrt{5}}{16}$

Step 5.3.2.6

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

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Step 5.3.2.6.1

Simplify the numerator.

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Step 5.3.2.6.1.1

Raise $- 24$ to the power of $2$ .

$x = \frac{24 \pm \sqrt{576 - 4 \cdot 64 \cdot 1}}{2 \cdot 64}$

Step 5.3.2.6.1.2

Multiply $- 4 \cdot 64 \cdot 1$ .

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Step 5.3.2.6.1.2.1

Multiply $- 4$ by $64$ .

$x = \frac{24 \pm \sqrt{576 - 256 \cdot 1}}{2 \cdot 64}$

Step 5.3.2.6.1.2.2

Multiply $- 256$ by $1$ .

$x = \frac{24 \pm \sqrt{576 - 256}}{2 \cdot 64}$

Step 5.3.2.6.1.3

Subtract $256$ from $576$ .

$x = \frac{24 \pm \sqrt{320}}{2 \cdot 64}$

Step 5.3.2.6.1.4

Rewrite $320$ as $8^{2} \cdot 5$ .

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Step 5.3.2.6.1.4.1

Factor $64$ out of $320$ .

$x = \frac{24 \pm \sqrt{64 (5)}}{2 \cdot 64}$

Step 5.3.2.6.1.4.2

Rewrite $64$ as $8^{2}$ .

$x = \frac{24 \pm \sqrt{8^{2} \cdot 5}}{2 \cdot 64}$

Step 5.3.2.6.1.5

Pull terms out from under the radical.

$x = \frac{24 \pm 8 \sqrt{5}}{2 \cdot 64}$

Step 5.3.2.6.2

Multiply $2$ by $64$ .

$x = \frac{24 \pm 8 \sqrt{5}}{128}$

Step 5.3.2.6.3

Simplify $\frac{24 \pm 8 \sqrt{5}}{128}$ .

$x = \frac{3 \pm \sqrt{5}}{16}$

Step 5.3.2.6.4

Change the $\pm$ to $-$ .

$x = \frac{3 - \sqrt{5}}{16}$

Step 5.3.2.7

Consolidate the solutions.

$x = \frac{3 + \sqrt{5}}{16}, \frac{3 - \sqrt{5}}{16}$

Step 5.3.2.8

Use each root to create test intervals.

$x < \frac{3 - \sqrt{5}}{16}$

$\frac{3 - \sqrt{5}}{16} < x < \frac{3 + \sqrt{5}}{16}$

$x > \frac{3 + \sqrt{5}}{16}$

Step 5.3.2.9

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 5.3.2.9.1

Test a value on the interval $x < \frac{3 - \sqrt{5}}{16}$ to see if it makes the inequality true.

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Step 5.3.2.9.1.1

Choose a value on the interval $x < \frac{3 - \sqrt{5}}{16}$ and see if this value makes the original inequality true.

$x = 0$

Step 5.3.2.9.1.2

Replace $x$ with $0$ in the original inequality.

$64 {(0)}^{2} - 24 \cdot 0 + 1 \geq 0$

Step 5.3.2.9.1.3

The left side $1$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 5.3.2.9.2

Test a value on the interval $\frac{3 - \sqrt{5}}{16} < x < \frac{3 + \sqrt{5}}{16}$ to see if it makes the inequality true.

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Step 5.3.2.9.2.1

Choose a value on the interval $\frac{3 - \sqrt{5}}{16} < x < \frac{3 + \sqrt{5}}{16}$ and see if this value makes the original inequality true.

$x = 0.19$

Step 5.3.2.9.2.2

Replace $x$ with $0.19$ in the original inequality.

$64 {(0.19)}^{2} - 24 \cdot 0.19 + 1 \geq 0$

Step 5.3.2.9.2.3

The left side $- 1.2496$ is less than the right side $0$ , which means that the given statement is false.

False

Step 5.3.2.9.3

Test a value on the interval $x > \frac{3 + \sqrt{5}}{16}$ to see if it makes the inequality true.

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Step 5.3.2.9.3.1

Choose a value on the interval $x > \frac{3 + \sqrt{5}}{16}$ and see if this value makes the original inequality true.

$x = 3$

Step 5.3.2.9.3.2

Replace $x$ with $3$ in the original inequality.

$64 {(3)}^{2} - 24 \cdot 3 + 1 \geq 0$

Step 5.3.2.9.3.3

The left side $505$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 5.3.2.9.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < \frac{3 - \sqrt{5}}{16}$ True

$\frac{3 - \sqrt{5}}{16} < x < \frac{3 + \sqrt{5}}{16}$ False

$x > \frac{3 + \sqrt{5}}{16}$ True

$x < \frac{3 - \sqrt{5}}{16}$ True

$\frac{3 - \sqrt{5}}{16} < x < \frac{3 + \sqrt{5}}{16}$ False

$x > \frac{3 + \sqrt{5}}{16}$ True

Step 5.3.2.10

The solution consists of all of the true intervals.

$x \leq \frac{3 - \sqrt{5}}{16}$ or $x \geq \frac{3 + \sqrt{5}}{16}$

Step 5.3.3

Set the denominator in $\frac{6 x + 1 + \sqrt{64 x^{2} - 24 x + 1}}{2 x}$ equal to $0$ to find where the expression is undefined.

$2 x = 0$

Step 5.3.4

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 5.3.4.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Step 5.3.4.2

Simplify the left side.

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Step 5.3.4.2.1

Cancel the common factor of $2$ .

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Step 5.3.4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Step 5.3.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 5.3.4.3

Simplify the right side.

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Step 5.3.4.3.1

Divide $0$ by $2$ .

$x = 0$

Step 5.3.5

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

$(- \infty, 0) \cup (0, \frac{3 - \sqrt{5}}{16}] \cup [\frac{3 + \sqrt{5}}{16}, \infty)$

Step 5.4

Since the domain of $f^{- 1} (x) = \frac{6 x + 1 + \sqrt{64 x^{2} - 24 x + 1}}{2 x}, \frac{6 x + 1 - \sqrt{64 x^{2} - 24 x + 1}}{2 x}$ is not equal to the range of $f (x) = \frac{x - 9}{(x - 7) (x + 1)}$ , then $f^{- 1} (x) = \frac{6 x + 1 + \sqrt{64 x^{2} - 24 x + 1}}{2 x}, \frac{6 x + 1 - \sqrt{64 x^{2} - 24 x + 1}}{2 x}$ is not an inverse of $f (x) = \frac{x - 9}{(x - 7) (x + 1)}$ .

There is no inverse

Step 6