Trigonometry Examples

$f (x) = \frac{x^{2} - 2 x - 35}{x^{2} + 9 x - 322}$

Step 1

Write $f (x) = \frac{x^{2} - 2 x - 35}{x^{2} + 9 x - 322}$ as an equation.

$y = \frac{x^{2} - 2 x - 35}{x^{2} + 9 x - 322}$

Step 2

Interchange the variables.

$x = \frac{y^{2} - 2 y - 35}{y^{2} + 9 y - 322}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{y^{2} - 2 y - 35}{y^{2} + 9 y - 322} = x$ .

$\frac{y^{2} - 2 y - 35}{y^{2} + 9 y - 322} = x$

Step 3.2

Factor each term.

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Step 3.2.1

Factor $y^{2} - 2 y - 35$ using the AC method.

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Step 3.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 35$ and whose sum is $- 2$ .

$- 7, 5$

Step 3.2.1.2

Write the factored form using these integers.

$\frac{(y - 7) (y + 5)}{y^{2} + 9 y - 322} = x$

Step 3.2.2

Factor $y^{2} + 9 y - 322$ using the AC method.

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Step 3.2.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 322$ and whose sum is $9$ .

$- 14, 23$

Step 3.2.2.2

Write the factored form using these integers.

$\frac{(y - 7) (y + 5)}{(y - 14) (y + 23)} = x$

Step 3.3

Find the LCD of the terms in the equation.

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Step 3.3.1

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

$(y - 14) (y + 23), 1$

Step 3.3.2

The LCM of one and any expression is the expression.

$(y - 14) (y + 23)$

Step 3.4

Multiply each term in $\frac{(y - 7) (y + 5)}{(y - 14) (y + 23)} = x$ by $(y - 14) (y + 23)$ to eliminate the fractions.

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Step 3.4.1

Multiply each term in $\frac{(y - 7) (y + 5)}{(y - 14) (y + 23)} = x$ by $(y - 14) (y + 23)$ .

$\frac{(y - 7) (y + 5)}{(y - 14) (y + 23)} ((y - 14) (y + 23)) = x ((y - 14) (y + 23))$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $(y - 14) (y + 23)$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{(y - 7) (y + 5)}{(y - 14) (y + 23)} ((y - 14) (y + 23)) = x ((y - 14) (y + 23))$

Step 3.4.2.1.2

Rewrite the expression.

$(y - 7) (y + 5) = x ((y - 14) (y + 23))$

Step 3.4.2.2

Expand $(y - 7) (y + 5)$ using the FOIL Method.

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Step 3.4.2.2.1

Apply the distributive property.

$y (y + 5) - 7 (y + 5) = x ((y - 14) (y + 23))$

Step 3.4.2.2.2

Apply the distributive property.

$y \cdot y + y \cdot 5 - 7 (y + 5) = x ((y - 14) (y + 23))$

Step 3.4.2.2.3

Apply the distributive property.

$y \cdot y + y \cdot 5 - 7 y - 7 \cdot 5 = x ((y - 14) (y + 23))$

Step 3.4.2.3

Simplify and combine like terms.

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Step 3.4.2.3.1

Simplify each term.

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Step 3.4.2.3.1.1

Multiply $y$ by $y$ .

$y^{2} + y \cdot 5 - 7 y - 7 \cdot 5 = x ((y - 14) (y + 23))$

Step 3.4.2.3.1.2

Move $5$ to the left of $y$ .

$y^{2} + 5 \cdot y - 7 y - 7 \cdot 5 = x ((y - 14) (y + 23))$

Step 3.4.2.3.1.3

Multiply $- 7$ by $5$ .

$y^{2} + 5 y - 7 y - 35 = x ((y - 14) (y + 23))$

Step 3.4.2.3.2

Subtract $7 y$ from $5 y$ .

$y^{2} - 2 y - 35 = x ((y - 14) (y + 23))$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Expand $(y - 14) (y + 23)$ using the FOIL Method.

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Step 3.4.3.1.1

Apply the distributive property.

$y^{2} - 2 y - 35 = x (y (y + 23) - 14 (y + 23))$

Step 3.4.3.1.2

Apply the distributive property.

$y^{2} - 2 y - 35 = x (y \cdot y + y \cdot 23 - 14 (y + 23))$

Step 3.4.3.1.3

Apply the distributive property.

$y^{2} - 2 y - 35 = x (y \cdot y + y \cdot 23 - 14 y - 14 \cdot 23)$

Step 3.4.3.2

Simplify and combine like terms.

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Step 3.4.3.2.1

Simplify each term.

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Step 3.4.3.2.1.1

Multiply $y$ by $y$ .

$y^{2} - 2 y - 35 = x (y^{2} + y \cdot 23 - 14 y - 14 \cdot 23)$

Step 3.4.3.2.1.2

Move $23$ to the left of $y$ .

$y^{2} - 2 y - 35 = x (y^{2} + 23 \cdot y - 14 y - 14 \cdot 23)$

Step 3.4.3.2.1.3

Multiply $- 14$ by $23$ .

$y^{2} - 2 y - 35 = x (y^{2} + 23 y - 14 y - 322)$

Step 3.4.3.2.2

Subtract $14 y$ from $23 y$ .

$y^{2} - 2 y - 35 = x (y^{2} + 9 y - 322)$

Step 3.4.3.3

Apply the distributive property.

$y^{2} - 2 y - 35 = x y^{2} + x (9 y) + x \cdot - 322$

Step 3.4.3.4

Simplify.

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Step 3.4.3.4.1

Rewrite using the commutative property of multiplication.

$y^{2} - 2 y - 35 = x y^{2} + 9 x y + x \cdot - 322$

Step 3.4.3.4.2

Move $- 322$ to the left of $x$ .

$y^{2} - 2 y - 35 = x y^{2} + 9 x y - 322 \cdot x$

$y^{2} - 2 y - 35 = x y^{2} + 9 x y - 322 x$

Step 3.5

Solve the equation.

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Step 3.5.1

Move all terms containing $y$ to the left side of the equation.

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Step 3.5.1.1

Subtract $x y^{2}$ from both sides of the equation.

$y^{2} - 2 y - 35 - x y^{2} = 9 x y - 322 x$

Step 3.5.1.2

Subtract $9 x y$ from both sides of the equation.

$y^{2} - 2 y - 35 - x y^{2} - 9 x y = - 322 x$

Step 3.5.2

Add $322 x$ to both sides of the equation.

$y^{2} - 2 y - 35 - x y^{2} - 9 x y + 322 x = 0$

Step 3.5.3

Use the quadratic formula to find the solutions.

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

Step 3.5.4

Substitute the values $a = 1 - x$ , $b = - 2 - 9 x$ , and $c = - 35 + 322 x$ into the quadratic formula and solve for $y$ .

$\frac{- (- 2 - 9 x) \pm \sqrt{{(- 2 - 9 x)}^{2} - 4 \cdot ((1 - x) \cdot (- 35 + 322 x))}}{2 (1 - x)}$

Step 3.5.5

Simplify the numerator.

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Step 3.5.5.1

Apply the distributive property.

$y = \frac{2 - (- 9 x) \pm \sqrt{{(- 2 - 9 x)}^{2} - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.5.2

Multiply $- 1$ by $- 2$ .

$y = \frac{2 - (- 9 x) \pm \sqrt{{(- 2 - 9 x)}^{2} - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.5.3

Multiply $- 9$ by $- 1$ .

$y = \frac{2 + 9 x \pm \sqrt{{(- 2 - 9 x)}^{2} - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.5.4

Rewrite ${(- 2 - 9 x)}^{2}$ as $(- 2 - 9 x) (- 2 - 9 x)$ .

$y = \frac{2 + 9 x \pm \sqrt{(- 2 - 9 x) (- 2 - 9 x) - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.5.5

Expand $(- 2 - 9 x) (- 2 - 9 x)$ using the FOIL Method.

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Step 3.5.5.5.1

Apply the distributive property.

$y = \frac{2 + 9 x \pm \sqrt{- 2 (- 2 - 9 x) - 9 x (- 2 - 9 x) - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.5.5.2

Apply the distributive property.

$y = \frac{2 + 9 x \pm \sqrt{- 2 \cdot - 2 - 2 (- 9 x) - 9 x (- 2 - 9 x) - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.5.5.3

Apply the distributive property.

$y = \frac{2 + 9 x \pm \sqrt{- 2 \cdot - 2 - 2 (- 9 x) - 9 x \cdot - 2 - 9 x (- 9 x) - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.5.6

Simplify and combine like terms.

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Step 3.5.5.6.1

Simplify each term.

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Step 3.5.5.6.1.1

Multiply $- 2$ by $- 2$ .

$y = \frac{2 + 9 x \pm \sqrt{4 - 2 (- 9 x) - 9 x \cdot - 2 - 9 x (- 9 x) - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.5.6.1.2

Multiply $- 9$ by $- 2$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 18 x - 9 x \cdot - 2 - 9 x (- 9 x) - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.5.6.1.3

Multiply $- 2$ by $- 9$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 18 x + 18 x - 9 x (- 9 x) - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.5.6.1.4

Rewrite using the commutative property of multiplication.

$y = \frac{2 + 9 x \pm \sqrt{4 + 18 x + 18 x - 9 \cdot (- 9 x \cdot x) - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.5.6.1.5

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

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Step 3.5.5.6.1.5.1

Move $x$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 18 x + 18 x - 9 \cdot (- 9 (x \cdot x)) - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.5.6.1.5.2

Multiply $x$ by $x$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 18 x + 18 x - 9 \cdot (- 9 x^{2}) - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.5.6.1.6

Multiply $- 9$ by $- 9$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 18 x + 18 x + 81 x^{2} - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.5.6.2

Add $18 x$ and $18 x$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.5.7

Apply the distributive property.

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} + (- 4 \cdot 1 - 4 (- x)) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.5.8

Multiply $- 4$ by $1$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} + (- 4 - 4 (- x)) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.5.9

Multiply $- 1$ by $- 4$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} + (- 4 + 4 x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.5.10

Expand $(- 4 + 4 x) (- 35 + 322 x)$ using the FOIL Method.

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Step 3.5.5.10.1

Apply the distributive property.

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} - 4 (- 35 + 322 x) + 4 x (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.5.10.2

Apply the distributive property.

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} - 4 \cdot - 35 - 4 (322 x) + 4 x (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.5.10.3

Apply the distributive property.

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} - 4 \cdot - 35 - 4 (322 x) + 4 x \cdot - 35 + 4 x (322 x)}}{2 (1 - x)}$

Step 3.5.5.11

Simplify and combine like terms.

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Step 3.5.5.11.1

Simplify each term.

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Step 3.5.5.11.1.1

Multiply $- 4$ by $- 35$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} + 140 - 4 (322 x) + 4 x \cdot - 35 + 4 x (322 x)}}{2 (1 - x)}$

Step 3.5.5.11.1.2

Multiply $322$ by $- 4$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} + 140 - 1288 x + 4 x \cdot - 35 + 4 x (322 x)}}{2 (1 - x)}$

Step 3.5.5.11.1.3

Multiply $- 35$ by $4$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} + 140 - 1288 x - 140 x + 4 x (322 x)}}{2 (1 - x)}$

Step 3.5.5.11.1.4

Rewrite using the commutative property of multiplication.

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} + 140 - 1288 x - 140 x + 4 \cdot (322 x \cdot x)}}{2 (1 - x)}$

Step 3.5.5.11.1.5

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

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Step 3.5.5.11.1.5.1

Move $x$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} + 140 - 1288 x - 140 x + 4 \cdot (322 (x \cdot x))}}{2 (1 - x)}$

Step 3.5.5.11.1.5.2

Multiply $x$ by $x$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} + 140 - 1288 x - 140 x + 4 \cdot (322 x^{2})}}{2 (1 - x)}$

Step 3.5.5.11.1.6

Multiply $4$ by $322$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} + 140 - 1288 x - 140 x + 1288 x^{2}}}{2 (1 - x)}$

Step 3.5.5.11.2

Subtract $140 x$ from $- 1288 x$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} + 140 - 1428 x + 1288 x^{2}}}{2 (1 - x)}$

Step 3.5.5.12

Add $4$ and $140$ .

$y = \frac{2 + 9 x \pm \sqrt{36 x + 81 x^{2} + 144 - 1428 x + 1288 x^{2}}}{2 (1 - x)}$

Step 3.5.5.13

Subtract $1428 x$ from $36 x$ .

$y = \frac{2 + 9 x \pm \sqrt{- 1392 x + 81 x^{2} + 144 + 1288 x^{2}}}{2 (1 - x)}$

Step 3.5.5.14

Add $81 x^{2}$ and $1288 x^{2}$ .

$y = \frac{2 + 9 x \pm \sqrt{- 1392 x + 1369 x^{2} + 144}}{2 (1 - x)}$

Step 3.5.5.15

Reorder terms.

$y = \frac{2 + 9 x \pm \sqrt{1369 x^{2} - 1392 x + 144}}{2 (1 - x)}$

Step 3.5.6

Change the $\pm$ to $+$ .

$y = \frac{2 + 9 x + \sqrt{1369 x^{2} - 1392 x + 144}}{2 (1 - x)}$

Step 3.5.7

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

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Step 3.5.7.1

Simplify the numerator.

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Step 3.5.7.1.1

Apply the distributive property.

$y = \frac{2 - (- 9 x) \pm \sqrt{{(- 2 - 9 x)}^{2} - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.7.1.2

Multiply $- 1$ by $- 2$ .

$y = \frac{2 - (- 9 x) \pm \sqrt{{(- 2 - 9 x)}^{2} - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.7.1.3

Multiply $- 9$ by $- 1$ .

$y = \frac{2 + 9 x \pm \sqrt{{(- 2 - 9 x)}^{2} - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.7.1.4

Rewrite ${(- 2 - 9 x)}^{2}$ as $(- 2 - 9 x) (- 2 - 9 x)$ .

$y = \frac{2 + 9 x \pm \sqrt{(- 2 - 9 x) (- 2 - 9 x) - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.7.1.5

Expand $(- 2 - 9 x) (- 2 - 9 x)$ using the FOIL Method.

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Step 3.5.7.1.5.1

Apply the distributive property.

$y = \frac{2 + 9 x \pm \sqrt{- 2 (- 2 - 9 x) - 9 x (- 2 - 9 x) - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.7.1.5.2

Apply the distributive property.

$y = \frac{2 + 9 x \pm \sqrt{- 2 \cdot - 2 - 2 (- 9 x) - 9 x (- 2 - 9 x) - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.7.1.5.3

Apply the distributive property.

$y = \frac{2 + 9 x \pm \sqrt{- 2 \cdot - 2 - 2 (- 9 x) - 9 x \cdot - 2 - 9 x (- 9 x) - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.7.1.6

Simplify and combine like terms.

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Step 3.5.7.1.6.1

Simplify each term.

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Step 3.5.7.1.6.1.1

Multiply $- 2$ by $- 2$ .

$y = \frac{2 + 9 x \pm \sqrt{4 - 2 (- 9 x) - 9 x \cdot - 2 - 9 x (- 9 x) - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.7.1.6.1.2

Multiply $- 9$ by $- 2$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 18 x - 9 x \cdot - 2 - 9 x (- 9 x) - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.7.1.6.1.3

Multiply $- 2$ by $- 9$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 18 x + 18 x - 9 x (- 9 x) - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.7.1.6.1.4

Rewrite using the commutative property of multiplication.

$y = \frac{2 + 9 x \pm \sqrt{4 + 18 x + 18 x - 9 \cdot (- 9 x \cdot x) - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.7.1.6.1.5

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

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Step 3.5.7.1.6.1.5.1

Move $x$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 18 x + 18 x - 9 \cdot (- 9 (x \cdot x)) - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.7.1.6.1.5.2

Multiply $x$ by $x$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 18 x + 18 x - 9 \cdot (- 9 x^{2}) - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.7.1.6.1.6

Multiply $- 9$ by $- 9$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 18 x + 18 x + 81 x^{2} - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.7.1.6.2

Add $18 x$ and $18 x$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} - 4 \cdot (1 - x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.7.1.7

Apply the distributive property.

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} + (- 4 \cdot 1 - 4 (- x)) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.7.1.8

Multiply $- 4$ by $1$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} + (- 4 - 4 (- x)) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.7.1.9

Multiply $- 1$ by $- 4$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} + (- 4 + 4 x) \cdot (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.7.1.10

Expand $(- 4 + 4 x) (- 35 + 322 x)$ using the FOIL Method.

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Step 3.5.7.1.10.1

Apply the distributive property.

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} - 4 (- 35 + 322 x) + 4 x (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.7.1.10.2

Apply the distributive property.

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} - 4 \cdot - 35 - 4 (322 x) + 4 x (- 35 + 322 x)}}{2 (1 - x)}$

Step 3.5.7.1.10.3

Apply the distributive property.

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} - 4 \cdot - 35 - 4 (322 x) + 4 x \cdot - 35 + 4 x (322 x)}}{2 (1 - x)}$

Step 3.5.7.1.11

Simplify and combine like terms.

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Step 3.5.7.1.11.1

Simplify each term.

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Step 3.5.7.1.11.1.1

Multiply $- 4$ by $- 35$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} + 140 - 4 (322 x) + 4 x \cdot - 35 + 4 x (322 x)}}{2 (1 - x)}$

Step 3.5.7.1.11.1.2

Multiply $322$ by $- 4$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} + 140 - 1288 x + 4 x \cdot - 35 + 4 x (322 x)}}{2 (1 - x)}$

Step 3.5.7.1.11.1.3

Multiply $- 35$ by $4$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} + 140 - 1288 x - 140 x + 4 x (322 x)}}{2 (1 - x)}$

Step 3.5.7.1.11.1.4

Rewrite using the commutative property of multiplication.

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} + 140 - 1288 x - 140 x + 4 \cdot (322 x \cdot x)}}{2 (1 - x)}$

Step 3.5.7.1.11.1.5

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

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Step 3.5.7.1.11.1.5.1

Move $x$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} + 140 - 1288 x - 140 x + 4 \cdot (322 (x \cdot x))}}{2 (1 - x)}$

Step 3.5.7.1.11.1.5.2

Multiply $x$ by $x$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} + 140 - 1288 x - 140 x + 4 \cdot (322 x^{2})}}{2 (1 - x)}$

Step 3.5.7.1.11.1.6

Multiply $4$ by $322$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} + 140 - 1288 x - 140 x + 1288 x^{2}}}{2 (1 - x)}$

Step 3.5.7.1.11.2

Subtract $140 x$ from $- 1288 x$ .

$y = \frac{2 + 9 x \pm \sqrt{4 + 36 x + 81 x^{2} + 140 - 1428 x + 1288 x^{2}}}{2 (1 - x)}$

Step 3.5.7.1.12

Add $4$ and $140$ .

$y = \frac{2 + 9 x \pm \sqrt{36 x + 81 x^{2} + 144 - 1428 x + 1288 x^{2}}}{2 (1 - x)}$

Step 3.5.7.1.13

Subtract $1428 x$ from $36 x$ .

$y = \frac{2 + 9 x \pm \sqrt{- 1392 x + 81 x^{2} + 144 + 1288 x^{2}}}{2 (1 - x)}$

Step 3.5.7.1.14

Add $81 x^{2}$ and $1288 x^{2}$ .

$y = \frac{2 + 9 x \pm \sqrt{- 1392 x + 1369 x^{2} + 144}}{2 (1 - x)}$

Step 3.5.7.1.15

Reorder terms.

$y = \frac{2 + 9 x \pm \sqrt{1369 x^{2} - 1392 x + 144}}{2 (1 - x)}$

Step 3.5.7.2

Change the $\pm$ to $-$ .

$y = \frac{2 + 9 x - \sqrt{1369 x^{2} - 1392 x + 144}}{2 (1 - x)}$

Step 3.5.8

The final answer is the combination of both solutions.

$y = \frac{2 + 9 x + \sqrt{1369 x^{2} - 1392 x + 144}}{2 (1 - x)}$

$y = \frac{2 + 9 x - \sqrt{1369 x^{2} - 1392 x + 144}}{2 (1 - x)}$

$y = \frac{2 + 9 x + \sqrt{1369 x^{2} - 1392 x + 144}}{2 (1 - x)}$

$y = \frac{2 + 9 x - \sqrt{1369 x^{2} - 1392 x + 144}}{2 (1 - x)}$

$y = \frac{2 + 9 x + \sqrt{1369 x^{2} - 1392 x + 144}}{2 (1 - x)}$

$y = \frac{2 + 9 x - \sqrt{1369 x^{2} - 1392 x + 144}}{2 (1 - x)}$

Step 4

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

$f^{- 1} (x) = \frac{2 + 9 x + \sqrt{1369 x^{2} - 1392 x + 144}}{2 (1 - x)}, \frac{2 + 9 x - \sqrt{1369 x^{2} - 1392 x + 144}}{2 (1 - x)}$

Step 5

Verify if $f^{- 1} (x) = \frac{2 + 9 x + \sqrt{1369 x^{2} - 1392 x + 144}}{2 (1 - x)}, \frac{2 + 9 x - \sqrt{1369 x^{2} - 1392 x + 144}}{2 (1 - x)}$ is the inverse of $f (x) = \frac{x^{2} - 2 x - 35}{x^{2} + 9 x - 322}$ .

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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 $f (x) = \frac{x^{2} - 2 x - 35}{x^{2} + 9 x - 322}$ and $f^{- 1} (x) = \frac{2 + 9 x + \sqrt{1369 x^{2} - 1392 x + 144}}{2 (1 - x)}, \frac{2 + 9 x - \sqrt{1369 x^{2} - 1392 x + 144}}{2 (1 - x)}$ and compare them.

Step 5.2

Find the range of $f (x) = \frac{x^{2} - 2 x - 35}{x^{2} + 9 x - 322}$ .

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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:

$(- \infty, \frac{696 - 12 \sqrt{1995}}{1369}] \cup [\frac{696 + 12 \sqrt{1995}}{1369}, \infty)$

Step 5.3

Find the domain of $\frac{2 + 9 x + \sqrt{1369 x^{2} - 1392 x + 144}}{2 (1 - x)}$ .

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Step 5.3.1

Set the radicand in $\sqrt{1369 x^{2} - 1392 x + 144}$ greater than or equal to $0$ to find where the expression is defined.

$1369 x^{2} - 1392 x + 144 \geq 0$

Step 5.3.2

Solve for $x$ .

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Step 5.3.2.1

Convert the inequality to an equation.

$1369 x^{2} - 1392 x + 144 = 0$

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 $a = 1369$ , $b = - 1392$ , and $c = 144$ into the quadratic formula and solve for $x$ .

$\frac{1392 \pm \sqrt{{(- 1392)}^{2} - 4 \cdot (1369 \cdot 144)}}{2 \cdot 1369}$

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 $- 1392$ to the power of $2$ .

$x = \frac{1392 \pm \sqrt{1937664 - 4 \cdot 1369 \cdot 144}}{2 \cdot 1369}$

Step 5.3.2.4.1.2

Multiply $- 4 \cdot 1369 \cdot 144$ .

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Step 5.3.2.4.1.2.1

Multiply $- 4$ by $1369$ .

$x = \frac{1392 \pm \sqrt{1937664 - 5476 \cdot 144}}{2 \cdot 1369}$

Step 5.3.2.4.1.2.2

Multiply $- 5476$ by $144$ .

$x = \frac{1392 \pm \sqrt{1937664 - 788544}}{2 \cdot 1369}$

Step 5.3.2.4.1.3

Subtract $788544$ from $1937664$ .

$x = \frac{1392 \pm \sqrt{1149120}}{2 \cdot 1369}$

Step 5.3.2.4.1.4

Rewrite $1149120$ as $24^{2} \cdot 1995$ .

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Step 5.3.2.4.1.4.1

Factor $576$ out of $1149120$ .

$x = \frac{1392 \pm \sqrt{576 (1995)}}{2 \cdot 1369}$

Step 5.3.2.4.1.4.2

Rewrite $576$ as $24^{2}$ .

$x = \frac{1392 \pm \sqrt{24^{2} \cdot 1995}}{2 \cdot 1369}$

Step 5.3.2.4.1.5

Pull terms out from under the radical.

$x = \frac{1392 \pm 24 \sqrt{1995}}{2 \cdot 1369}$

Step 5.3.2.4.2

Multiply $2$ by $1369$ .

$x = \frac{1392 \pm 24 \sqrt{1995}}{2738}$

Step 5.3.2.4.3

Simplify $\frac{1392 \pm 24 \sqrt{1995}}{2738}$ .

$x = \frac{696 \pm 12 \sqrt{1995}}{1369}$

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 $- 1392$ to the power of $2$ .

$x = \frac{1392 \pm \sqrt{1937664 - 4 \cdot 1369 \cdot 144}}{2 \cdot 1369}$

Step 5.3.2.5.1.2

Multiply $- 4 \cdot 1369 \cdot 144$ .

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Step 5.3.2.5.1.2.1

Multiply $- 4$ by $1369$ .

$x = \frac{1392 \pm \sqrt{1937664 - 5476 \cdot 144}}{2 \cdot 1369}$

Step 5.3.2.5.1.2.2

Multiply $- 5476$ by $144$ .

$x = \frac{1392 \pm \sqrt{1937664 - 788544}}{2 \cdot 1369}$

Step 5.3.2.5.1.3

Subtract $788544$ from $1937664$ .

$x = \frac{1392 \pm \sqrt{1149120}}{2 \cdot 1369}$

Step 5.3.2.5.1.4

Rewrite $1149120$ as $24^{2} \cdot 1995$ .

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Step 5.3.2.5.1.4.1

Factor $576$ out of $1149120$ .

$x = \frac{1392 \pm \sqrt{576 (1995)}}{2 \cdot 1369}$

Step 5.3.2.5.1.4.2

Rewrite $576$ as $24^{2}$ .

$x = \frac{1392 \pm \sqrt{24^{2} \cdot 1995}}{2 \cdot 1369}$

Step 5.3.2.5.1.5

Pull terms out from under the radical.

$x = \frac{1392 \pm 24 \sqrt{1995}}{2 \cdot 1369}$

Step 5.3.2.5.2

Multiply $2$ by $1369$ .

$x = \frac{1392 \pm 24 \sqrt{1995}}{2738}$

Step 5.3.2.5.3

Simplify $\frac{1392 \pm 24 \sqrt{1995}}{2738}$ .

$x = \frac{696 \pm 12 \sqrt{1995}}{1369}$

Step 5.3.2.5.4

Change the $\pm$ to $+$ .

$x = \frac{696 + 12 \sqrt{1995}}{1369}$

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 $- 1392$ to the power of $2$ .

$x = \frac{1392 \pm \sqrt{1937664 - 4 \cdot 1369 \cdot 144}}{2 \cdot 1369}$

Step 5.3.2.6.1.2

Multiply $- 4 \cdot 1369 \cdot 144$ .

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Step 5.3.2.6.1.2.1

Multiply $- 4$ by $1369$ .

$x = \frac{1392 \pm \sqrt{1937664 - 5476 \cdot 144}}{2 \cdot 1369}$

Step 5.3.2.6.1.2.2

Multiply $- 5476$ by $144$ .

$x = \frac{1392 \pm \sqrt{1937664 - 788544}}{2 \cdot 1369}$

Step 5.3.2.6.1.3

Subtract $788544$ from $1937664$ .

$x = \frac{1392 \pm \sqrt{1149120}}{2 \cdot 1369}$

Step 5.3.2.6.1.4

Rewrite $1149120$ as $24^{2} \cdot 1995$ .

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Step 5.3.2.6.1.4.1

Factor $576$ out of $1149120$ .

$x = \frac{1392 \pm \sqrt{576 (1995)}}{2 \cdot 1369}$

Step 5.3.2.6.1.4.2

Rewrite $576$ as $24^{2}$ .

$x = \frac{1392 \pm \sqrt{24^{2} \cdot 1995}}{2 \cdot 1369}$

Step 5.3.2.6.1.5

Pull terms out from under the radical.

$x = \frac{1392 \pm 24 \sqrt{1995}}{2 \cdot 1369}$

Step 5.3.2.6.2

Multiply $2$ by $1369$ .

$x = \frac{1392 \pm 24 \sqrt{1995}}{2738}$

Step 5.3.2.6.3

Simplify $\frac{1392 \pm 24 \sqrt{1995}}{2738}$ .

$x = \frac{696 \pm 12 \sqrt{1995}}{1369}$

Step 5.3.2.6.4

Change the $\pm$ to $-$ .

$x = \frac{696 - 12 \sqrt{1995}}{1369}$

Step 5.3.2.7

Consolidate the solutions.

$x = \frac{696 + 12 \sqrt{1995}}{1369}, \frac{696 - 12 \sqrt{1995}}{1369}$

Step 5.3.2.8

Use each root to create test intervals.

$x < \frac{696 - 12 \sqrt{1995}}{1369}$

$\frac{696 - 12 \sqrt{1995}}{1369} < x < \frac{696 + 12 \sqrt{1995}}{1369}$

$x > \frac{696 + 12 \sqrt{1995}}{1369}$

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 $x < \frac{696 - 12 \sqrt{1995}}{1369}$ 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 $x < \frac{696 - 12 \sqrt{1995}}{1369}$ 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.

$1369 {(0)}^{2} - 1392 \cdot 0 + 144 \geq 0$

Step 5.3.2.9.1.3

The left side $144$ 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 $\frac{696 - 12 \sqrt{1995}}{1369} < x < \frac{696 + 12 \sqrt{1995}}{1369}$ 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 $\frac{696 - 12 \sqrt{1995}}{1369} < x < \frac{696 + 12 \sqrt{1995}}{1369}$ and see if this value makes the original inequality true.

$x = 0.51$

Step 5.3.2.9.2.2

Replace $x$ with $0.51$ in the original inequality.

$1369 {(0.51)}^{2} - 1392 \cdot 0.51 + 144 \geq 0$

Step 5.3.2.9.2.3

The left side $- 209.8431$ 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 $x > \frac{696 + 12 \sqrt{1995}}{1369}$ 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 $x > \frac{696 + 12 \sqrt{1995}}{1369}$ 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.

$1369 {(3)}^{2} - 1392 \cdot 3 + 144 \geq 0$

Step 5.3.2.9.3.3

The left side $8289$ 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.

$x < \frac{696 - 12 \sqrt{1995}}{1369}$ True

$\frac{696 - 12 \sqrt{1995}}{1369} < x < \frac{696 + 12 \sqrt{1995}}{1369}$ False

$x > \frac{696 + 12 \sqrt{1995}}{1369}$ True

$x < \frac{696 - 12 \sqrt{1995}}{1369}$ True

$\frac{696 - 12 \sqrt{1995}}{1369} < x < \frac{696 + 12 \sqrt{1995}}{1369}$ False

$x > \frac{696 + 12 \sqrt{1995}}{1369}$ True

Step 5.3.2.10

The solution consists of all of the true intervals.

$x \leq \frac{696 - 12 \sqrt{1995}}{1369}$ or $x \geq \frac{696 + 12 \sqrt{1995}}{1369}$

Step 5.3.3

Set the denominator in $\frac{2 + 9 x + \sqrt{1369 x^{2} - 1392 x + 144}}{2 (1 - x)}$ equal to $0$ to find where the expression is undefined.

$2 (1 - x) = 0$

Step 5.3.4

Solve for $x$ .

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Step 5.3.4.1

Divide each term in $2 (1 - x) = 0$ by $2$ and simplify.

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Step 5.3.4.1.1

Divide each term in $2 (1 - x) = 0$ by $2$ .

$\frac{2 (1 - x)}{2} = \frac{0}{2}$

Step 5.3.4.1.2

Simplify the left side.

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Step 5.3.4.1.2.1

Cancel the common factor of $2$ .

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Step 5.3.4.1.2.1.1

Cancel the common factor.

$\frac{2 (1 - x)}{2} = \frac{0}{2}$

Step 5.3.4.1.2.1.2

Divide $1 - x$ by $1$ .

$1 - x = \frac{0}{2}$

Step 5.3.4.1.3

Simplify the right side.

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Step 5.3.4.1.3.1

Divide $0$ by $2$ .

$1 - x = 0$

Step 5.3.4.2

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 5.3.4.3

Divide each term in $- x = - 1$ by $- 1$ and simplify.

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Step 5.3.4.3.1

Divide each term in $- x = - 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 1}{- 1}$

Step 5.3.4.3.2

Simplify the left side.

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Step 5.3.4.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 1}{- 1}$

Step 5.3.4.3.2.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 1}$

Step 5.3.4.3.3

Simplify the right side.

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Step 5.3.4.3.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 5.3.5

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

$(- \infty, \frac{696 - 12 \sqrt{1995}}{1369}] \cup [\frac{696 + 12 \sqrt{1995}}{1369}, 1) \cup (1, \infty)$

Step 5.4

Since the domain of $f^{- 1} (x) = \frac{2 + 9 x + \sqrt{1369 x^{2} - 1392 x + 144}}{2 (1 - x)}, \frac{2 + 9 x - \sqrt{1369 x^{2} - 1392 x + 144}}{2 (1 - x)}$ is not equal to the range of $f (x) = \frac{x^{2} - 2 x - 35}{x^{2} + 9 x - 322}$ , then $f^{- 1} (x) = \frac{2 + 9 x + \sqrt{1369 x^{2} - 1392 x + 144}}{2 (1 - x)}, \frac{2 + 9 x - \sqrt{1369 x^{2} - 1392 x + 144}}{2 (1 - x)}$ is not an inverse of $f (x) = \frac{x^{2} - 2 x - 35}{x^{2} + 9 x - 322}$ .

There is no inverse

Step 6