Precalculus Examples

$f (x) = \frac{x^{2} + 2 x + 1}{x^{2} + x - 2}$

Step 1

Write $f (x) = \frac{x^{2} + 2 x + 1}{x^{2} + x - 2}$ as an equation.

$y = \frac{x^{2} + 2 x + 1}{x^{2} + x - 2}$

Step 2

Interchange the variables.

$x = \frac{y^{2} + 2 y + 1}{y^{2} + y - 2}$

Step 3

Solve for $y$ .

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Step 3.1

Multiply the equation by $y^{2} + y - 2$ .

$(y^{2} + y - 2) (x) = (\frac{y^{2} + 2 y + 1}{y^{2} + y - 2}) (y^{2} + y - 2)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Apply the distributive property.

$y^{2} x + y x - 2 x = (\frac{y^{2} + 2 y + 1}{y^{2} + y - 2}) (y^{2} + y - 2)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify $(\frac{y^{2} + 2 y + 1}{y^{2} + y - 2}) (y^{2} + y - 2)$ .

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Step 3.3.1.1

Factor using the perfect square rule.

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Step 3.3.1.1.1

Rewrite $1$ as $1^{2}$ .

$y^{2} x + y x - 2 x = \frac{y^{2} + 2 y + 1^{2}}{y^{2} + y - 2} (y^{2} + y - 2)$

Step 3.3.1.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 y = 2 \cdot y \cdot 1$

Step 3.3.1.1.3

Rewrite the polynomial.

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

Step 3.3.1.1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = y$ and $b = 1$ .

$y^{2} x + y x - 2 x = \frac{{(y + 1)}^{2}}{y^{2} + y - 2} (y^{2} + y - 2)$

Step 3.3.1.2

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

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Step 3.3.1.2.1

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 $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 3.3.1.2.2

Write the factored form using these integers.

$y^{2} x + y x - 2 x = \frac{{(y + 1)}^{2}}{(y - 1) (y + 2)} (y^{2} + y - 2)$

Step 3.3.1.3

Multiply $\frac{{(y + 1)}^{2}}{(y - 1) (y + 2)}$ by $y^{2} + y - 2$ .

$y^{2} x + y x - 2 x = \frac{{(y + 1)}^{2} (y^{2} + y - 2)}{(y - 1) (y + 2)}$

Step 3.3.1.4

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

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Step 3.3.1.4.1

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 $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 3.3.1.4.2

Write the factored form using these integers.

$y^{2} x + y x - 2 x = \frac{{(y + 1)}^{2} ((y - 1) (y + 2))}{(y - 1) (y + 2)}$

$y^{2} x + y x - 2 x = \frac{{(y + 1)}^{2} (y - 1) (y + 2)}{(y - 1) (y + 2)}$

Step 3.3.1.5

Reduce the expression by cancelling the common factors.

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Step 3.3.1.5.1

Cancel the common factor of $y - 1$ .

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Step 3.3.1.5.1.1

Cancel the common factor.

$y^{2} x + y x - 2 x = \frac{{(y + 1)}^{2} (y - 1) (y + 2)}{(y - 1) (y + 2)}$

Step 3.3.1.5.1.2

Rewrite the expression.

$y^{2} x + y x - 2 x = \frac{{(y + 1)}^{2} (y + 2)}{y + 2}$

Step 3.3.1.5.2

Cancel the common factor of $y + 2$ .

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Step 3.3.1.5.2.1

Cancel the common factor.

$y^{2} x + y x - 2 x = \frac{{(y + 1)}^{2} (y + 2)}{y + 2}$

Step 3.3.1.5.2.2

Divide ${(y + 1)}^{2}$ by $1$ .

$y^{2} x + y x - 2 x = {(y + 1)}^{2}$

Step 3.3.1.5.3

Rewrite ${(y + 1)}^{2}$ as $(y + 1) (y + 1)$ .

$y^{2} x + y x - 2 x = (y + 1) (y + 1)$

Step 3.3.1.6

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

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Step 3.3.1.6.1

Apply the distributive property.

$y^{2} x + y x - 2 x = y (y + 1) + 1 (y + 1)$

Step 3.3.1.6.2

Apply the distributive property.

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

Step 3.3.1.6.3

Apply the distributive property.

$y^{2} x + y x - 2 x = y \cdot y + y \cdot 1 + 1 y + 1 \cdot 1$

Step 3.3.1.7

Simplify and combine like terms.

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Step 3.3.1.7.1

Simplify each term.

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Step 3.3.1.7.1.1

Multiply $y$ by $y$ .

$y^{2} x + y x - 2 x = y^{2} + y \cdot 1 + 1 y + 1 \cdot 1$

Step 3.3.1.7.1.2

Multiply $y$ by $1$ .

$y^{2} x + y x - 2 x = y^{2} + y + 1 y + 1 \cdot 1$

Step 3.3.1.7.1.3

Multiply $y$ by $1$ .

$y^{2} x + y x - 2 x = y^{2} + y + y + 1 \cdot 1$

Step 3.3.1.7.1.4

Multiply $1$ by $1$ .

$y^{2} x + y x - 2 x = y^{2} + y + y + 1$

Step 3.3.1.7.2

Add $y$ and $y$ .

$y^{2} x + y x - 2 x = y^{2} + 2 y + 1$

Step 3.4

Solve for $y$ .

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Step 3.4.1

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

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Step 3.4.1.1

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

$y^{2} x + y x - 2 x - y^{2} = 2 y + 1$

Step 3.4.1.2

Subtract $2 y$ from both sides of the equation.

$y^{2} x + y x - 2 x - y^{2} - 2 y = 1$

Step 3.4.2

Subtract $1$ from both sides of the equation.

$y^{2} x + y x - 2 x - y^{2} - 2 y - 1 = 0$

Step 3.4.3

Use the quadratic formula to find the solutions.

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

Step 3.4.4

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

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

Step 3.4.5

Simplify the numerator.

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Step 3.4.5.1

Apply the distributive property.

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

Step 3.4.5.2

Multiply $- 1$ by $- 2$ .

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

Step 3.4.5.3

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

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

Step 3.4.5.4

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

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Step 3.4.5.4.1

Apply the distributive property.

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

Step 3.4.5.4.2

Apply the distributive property.

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

Step 3.4.5.4.3

Apply the distributive property.

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

Step 3.4.5.5

Simplify and combine like terms.

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Step 3.4.5.5.1

Simplify each term.

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Step 3.4.5.5.1.1

Multiply $x$ by $x$ .

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

Step 3.4.5.5.1.2

Move $- 2$ to the left of $x$ .

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

Step 3.4.5.5.1.3

Multiply $- 2$ by $- 2$ .

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

Step 3.4.5.5.2

Subtract $2 x$ from $- 2 x$ .

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

Step 3.4.5.6

Apply the distributive property.

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

Step 3.4.5.7

Multiply $- 4$ by $- 1$ .

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

Step 3.4.5.8

Expand $(- 4 x + 4) (- 2 x - 1)$ using the FOIL Method.

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Step 3.4.5.8.1

Apply the distributive property.

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

Step 3.4.5.8.2

Apply the distributive property.

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

Step 3.4.5.8.3

Apply the distributive property.

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

Step 3.4.5.9

Simplify and combine like terms.

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Step 3.4.5.9.1

Simplify each term.

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Step 3.4.5.9.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.4.5.9.1.2

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

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Step 3.4.5.9.1.2.1

Move $x$ .

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

Step 3.4.5.9.1.2.2

Multiply $x$ by $x$ .

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

Step 3.4.5.9.1.3

Multiply $- 4$ by $- 2$ .

$y = \frac{- x + 2 \pm \sqrt{x^{2} - 4 x + 4 + 8 x^{2} - 4 x \cdot - 1 + 4 (- 2 x) + 4 \cdot - 1}}{2 (x - 1)}$

Step 3.4.5.9.1.4

Multiply $- 1$ by $- 4$ .

$y = \frac{- x + 2 \pm \sqrt{x^{2} - 4 x + 4 + 8 x^{2} + 4 x + 4 (- 2 x) + 4 \cdot - 1}}{2 (x - 1)}$

Step 3.4.5.9.1.5

Multiply $- 2$ by $4$ .

$y = \frac{- x + 2 \pm \sqrt{x^{2} - 4 x + 4 + 8 x^{2} + 4 x - 8 x + 4 \cdot - 1}}{2 (x - 1)}$

Step 3.4.5.9.1.6

Multiply $4$ by $- 1$ .

$y = \frac{- x + 2 \pm \sqrt{x^{2} - 4 x + 4 + 8 x^{2} + 4 x - 8 x - 4}}{2 (x - 1)}$

Step 3.4.5.9.2

Subtract $8 x$ from $4 x$ .

$y = \frac{- x + 2 \pm \sqrt{x^{2} - 4 x + 4 + 8 x^{2} - 4 x - 4}}{2 (x - 1)}$

Step 3.4.5.10

Add $x^{2}$ and $8 x^{2}$ .

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

Step 3.4.5.11

Subtract $4 x$ from $- 4 x$ .

$y = \frac{- x + 2 \pm \sqrt{9 x^{2} - 8 x + 4 - 4}}{2 (x - 1)}$

Step 3.4.5.12

Subtract $4$ from $4$ .

$y = \frac{- x + 2 \pm \sqrt{9 x^{2} - 8 x + 0}}{2 (x - 1)}$

Step 3.4.5.13

Add $9 x^{2} - 8 x$ and $0$ .

$y = \frac{- x + 2 \pm \sqrt{9 x^{2} - 8 x}}{2 (x - 1)}$

Step 3.4.5.14

Factor $x$ out of $9 x^{2} - 8 x$ .

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Step 3.4.5.14.1

Factor $x$ out of $9 x^{2}$ .

$y = \frac{- x + 2 \pm \sqrt{x (9 x) - 8 x}}{2 (x - 1)}$

Step 3.4.5.14.2

Factor $x$ out of $- 8 x$ .

$y = \frac{- x + 2 \pm \sqrt{x (9 x) + x \cdot - 8}}{2 (x - 1)}$

Step 3.4.5.14.3

Factor $x$ out of $x (9 x) + x \cdot - 8$ .

$y = \frac{- x + 2 \pm \sqrt{x (9 x - 8)}}{2 (x - 1)}$

Step 3.4.6

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

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Step 3.4.6.1

Change the $\pm$ to $+$ .

$y = \frac{- x + 2 + \sqrt{x (9 x - 8)}}{2 (x - 1)}$

Step 3.4.6.2

Factor $- 1$ out of $- x$ .

$y = \frac{- (x) + 2 + \sqrt{x (9 x - 8)}}{2 (x - 1)}$

Step 3.4.6.3

Rewrite $2$ as $- 1 (- 2)$ .

$y = \frac{- (x) - 1 \cdot - 2 + \sqrt{x (9 x - 8)}}{2 (x - 1)}$

Step 3.4.6.4

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

$y = \frac{- (x - 2) + \sqrt{x (9 x - 8)}}{2 (x - 1)}$

Step 3.4.6.5

Factor $- 1$ out of $\sqrt{x (9 x - 8)}$ .

$y = \frac{- (x - 2) - 1 (- \sqrt{x (9 x - 8)})}{2 (x - 1)}$

Step 3.4.6.6

Factor $- 1$ out of $- (x - 2) - 1 (- \sqrt{x (9 x - 8)})$ .

$y = \frac{- (x - 2 - \sqrt{x (9 x - 8)})}{2 (x - 1)}$

Step 3.4.6.7

Rewrite $- (x - 2 - \sqrt{x (9 x - 8)})$ as $- 1 (x - 2 - \sqrt{x (9 x - 8)})$ .

$y = \frac{- 1 (x - 2 - \sqrt{x (9 x - 8)})}{2 (x - 1)}$

Step 3.4.6.8

Move the negative in front of the fraction.

$y = - \frac{x - 2 - \sqrt{x (9 x - 8)}}{2 (x - 1)}$

Step 3.4.7

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

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Step 3.4.7.1

Simplify the numerator.

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Step 3.4.7.1.1

Apply the distributive property.

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

Step 3.4.7.1.2

Multiply $- 1$ by $- 2$ .

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

Step 3.4.7.1.3

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

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

Step 3.4.7.1.4

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

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Step 3.4.7.1.4.1

Apply the distributive property.

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

Step 3.4.7.1.4.2

Apply the distributive property.

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

Step 3.4.7.1.4.3

Apply the distributive property.

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

Step 3.4.7.1.5

Simplify and combine like terms.

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Step 3.4.7.1.5.1

Simplify each term.

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Step 3.4.7.1.5.1.1

Multiply $x$ by $x$ .

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

Step 3.4.7.1.5.1.2

Move $- 2$ to the left of $x$ .

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

Step 3.4.7.1.5.1.3

Multiply $- 2$ by $- 2$ .

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

Step 3.4.7.1.5.2

Subtract $2 x$ from $- 2 x$ .

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

Step 3.4.7.1.6

Apply the distributive property.

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

Step 3.4.7.1.7

Multiply $- 4$ by $- 1$ .

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

Step 3.4.7.1.8

Expand $(- 4 x + 4) (- 2 x - 1)$ using the FOIL Method.

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Step 3.4.7.1.8.1

Apply the distributive property.

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

Step 3.4.7.1.8.2

Apply the distributive property.

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

Step 3.4.7.1.8.3

Apply the distributive property.

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

Step 3.4.7.1.9

Simplify and combine like terms.

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Step 3.4.7.1.9.1

Simplify each term.

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Step 3.4.7.1.9.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.4.7.1.9.1.2

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

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Step 3.4.7.1.9.1.2.1

Move $x$ .

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

Step 3.4.7.1.9.1.2.2

Multiply $x$ by $x$ .

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

Step 3.4.7.1.9.1.3

Multiply $- 4$ by $- 2$ .

$y = \frac{- x + 2 \pm \sqrt{x^{2} - 4 x + 4 + 8 x^{2} - 4 x \cdot - 1 + 4 (- 2 x) + 4 \cdot - 1}}{2 (x - 1)}$

Step 3.4.7.1.9.1.4

Multiply $- 1$ by $- 4$ .

$y = \frac{- x + 2 \pm \sqrt{x^{2} - 4 x + 4 + 8 x^{2} + 4 x + 4 (- 2 x) + 4 \cdot - 1}}{2 (x - 1)}$

Step 3.4.7.1.9.1.5

Multiply $- 2$ by $4$ .

$y = \frac{- x + 2 \pm \sqrt{x^{2} - 4 x + 4 + 8 x^{2} + 4 x - 8 x + 4 \cdot - 1}}{2 (x - 1)}$

Step 3.4.7.1.9.1.6

Multiply $4$ by $- 1$ .

$y = \frac{- x + 2 \pm \sqrt{x^{2} - 4 x + 4 + 8 x^{2} + 4 x - 8 x - 4}}{2 (x - 1)}$

Step 3.4.7.1.9.2

Subtract $8 x$ from $4 x$ .

$y = \frac{- x + 2 \pm \sqrt{x^{2} - 4 x + 4 + 8 x^{2} - 4 x - 4}}{2 (x - 1)}$

Step 3.4.7.1.10

Add $x^{2}$ and $8 x^{2}$ .

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

Step 3.4.7.1.11

Subtract $4 x$ from $- 4 x$ .

$y = \frac{- x + 2 \pm \sqrt{9 x^{2} - 8 x + 4 - 4}}{2 (x - 1)}$

Step 3.4.7.1.12

Subtract $4$ from $4$ .

$y = \frac{- x + 2 \pm \sqrt{9 x^{2} - 8 x + 0}}{2 (x - 1)}$

Step 3.4.7.1.13

Add $9 x^{2} - 8 x$ and $0$ .

$y = \frac{- x + 2 \pm \sqrt{9 x^{2} - 8 x}}{2 (x - 1)}$

Step 3.4.7.1.14

Factor $x$ out of $9 x^{2} - 8 x$ .

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Step 3.4.7.1.14.1

Factor $x$ out of $9 x^{2}$ .

$y = \frac{- x + 2 \pm \sqrt{x (9 x) - 8 x}}{2 (x - 1)}$

Step 3.4.7.1.14.2

Factor $x$ out of $- 8 x$ .

$y = \frac{- x + 2 \pm \sqrt{x (9 x) + x \cdot - 8}}{2 (x - 1)}$

Step 3.4.7.1.14.3

Factor $x$ out of $x (9 x) + x \cdot - 8$ .

$y = \frac{- x + 2 \pm \sqrt{x (9 x - 8)}}{2 (x - 1)}$

Step 3.4.7.2

Change the $\pm$ to $-$ .

$y = \frac{- x + 2 - \sqrt{x (9 x - 8)}}{2 (x - 1)}$

Step 3.4.7.3

Factor $- 1$ out of $- x$ .

$y = \frac{- (x) + 2 - \sqrt{x (9 x - 8)}}{2 (x - 1)}$

Step 3.4.7.4

Rewrite $2$ as $- 1 (- 2)$ .

$y = \frac{- (x) - 1 \cdot - 2 - \sqrt{x (9 x - 8)}}{2 (x - 1)}$

Step 3.4.7.5

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

$y = \frac{- (x - 2) - \sqrt{x (9 x - 8)}}{2 (x - 1)}$

Step 3.4.7.6

Factor $- 1$ out of $- \sqrt{x (9 x - 8)}$ .

$y = \frac{- (x - 2) - (\sqrt{x (9 x - 8)})}{2 (x - 1)}$

Step 3.4.7.7

Factor $- 1$ out of $- (x - 2) - (\sqrt{x (9 x - 8)})$ .

$y = \frac{- (x - 2 + \sqrt{x (9 x - 8)})}{2 (x - 1)}$

Step 3.4.7.8

Rewrite $- (x - 2 + \sqrt{x (9 x - 8)})$ as $- 1 (x - 2 + \sqrt{x (9 x - 8)})$ .

$y = \frac{- 1 (x - 2 + \sqrt{x (9 x - 8)})}{2 (x - 1)}$

Step 3.4.7.9

Move the negative in front of the fraction.

$y = - \frac{x - 2 + \sqrt{x (9 x - 8)}}{2 (x - 1)}$

Step 3.4.8

The final answer is the combination of both solutions.

$y = - \frac{x - 2 - \sqrt{x (9 x - 8)}}{2 (x - 1)}$

$y = - \frac{x - 2 + \sqrt{x (9 x - 8)}}{2 (x - 1)}$

$y = - \frac{x - 2 - \sqrt{x (9 x - 8)}}{2 (x - 1)}$

$y = - \frac{x - 2 + \sqrt{x (9 x - 8)}}{2 (x - 1)}$

$y = - \frac{x - 2 - \sqrt{x (9 x - 8)}}{2 (x - 1)}$

$y = - \frac{x - 2 + \sqrt{x (9 x - 8)}}{2 (x - 1)}$

Step 4

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

$f^{- 1} (x) = - \frac{x - 2 - \sqrt{x (9 x - 8)}}{2 (x - 1)}, - \frac{x - 2 + \sqrt{x (9 x - 8)}}{2 (x - 1)}$

Step 5

Verify if $f^{- 1} (x) = - \frac{x - 2 - \sqrt{x (9 x - 8)}}{2 (x - 1)}, - \frac{x - 2 + \sqrt{x (9 x - 8)}}{2 (x - 1)}$ is the inverse of $f (x) = \frac{x^{2} + 2 x + 1}{x^{2} + x - 2}$ .

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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 $f (x) = \frac{x^{2} + 2 x + 1}{x^{2} + x - 2}$ and $f^{- 1} (x) = - \frac{x - 2 - \sqrt{x (9 x - 8)}}{2 (x - 1)}, - \frac{x - 2 + \sqrt{x (9 x - 8)}}{2 (x - 1)}$ and compare them.

Step 5.2

Find the range of $f (x) = \frac{x^{2} + 2 x + 1}{x^{2} + x - 2}$ .

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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:

$(- \infty, 0] \cup [\frac{8}{9}, \infty)$

Step 5.3

Find the domain of $- \frac{x - 2 - \sqrt{x (9 x - 8)}}{2 (x - 1)}$ .

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Step 5.3.1

Set the radicand in $\sqrt{x (9 x - 8)}$ greater than or equal to $0$ to find where the expression is defined.

$x (9 x - 8) \geq 0$

Step 5.3.2

Solve for $x$ .

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Step 5.3.2.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$9 x - 8 = 0$

Step 5.3.2.2

Set $x$ equal to $0$ .

$x = 0$

Step 5.3.2.3

Set $9 x - 8$ equal to $0$ and solve for $x$ .

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Step 5.3.2.3.1

Set $9 x - 8$ equal to $0$ .

$9 x - 8 = 0$

Step 5.3.2.3.2

Solve $9 x - 8 = 0$ for $x$ .

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Step 5.3.2.3.2.1

Add $8$ to both sides of the equation.

$9 x = 8$

Step 5.3.2.3.2.2

Divide each term in $9 x = 8$ by $9$ and simplify.

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Step 5.3.2.3.2.2.1

Divide each term in $9 x = 8$ by $9$ .

$\frac{9 x}{9} = \frac{8}{9}$

Step 5.3.2.3.2.2.2

Simplify the left side.

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Step 5.3.2.3.2.2.2.1

Cancel the common factor of $9$ .

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Step 5.3.2.3.2.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{8}{9}$

Step 5.3.2.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{8}{9}$

Step 5.3.2.4

The final solution is all the values that make $x (9 x - 8) \geq 0$ true.

$x = 0, \frac{8}{9}$

Step 5.3.2.5

Use each root to create test intervals.

$x < 0$

$0 < x < \frac{8}{9}$

$x > \frac{8}{9}$

Step 5.3.2.6

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 5.3.2.6.1

Test a value on the interval $x < 0$ to see if it makes the inequality true.

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Step 5.3.2.6.1.1

Choose a value on the interval $x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 5.3.2.6.1.2

Replace $x$ with $- 2$ in the original inequality.

$(- 2) (9 (- 2) - 8) \geq 0$

Step 5.3.2.6.1.3

The left side $52$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 5.3.2.6.2

Test a value on the interval $0 < x < \frac{8}{9}$ to see if it makes the inequality true.

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Step 5.3.2.6.2.1

Choose a value on the interval $0 < x < \frac{8}{9}$ and see if this value makes the original inequality true.

$x = 0.44$

Step 5.3.2.6.2.2

Replace $x$ with $0.44$ in the original inequality.

$(0.44) (9 (0.44) - 8) \geq 0$

Step 5.3.2.6.2.3

The left side $- 1.7776$ is less than the right side $0$ , which means that the given statement is false.

False

Step 5.3.2.6.3

Test a value on the interval $x > \frac{8}{9}$ to see if it makes the inequality true.

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Step 5.3.2.6.3.1

Choose a value on the interval $x > \frac{8}{9}$ and see if this value makes the original inequality true.

$x = 3$

Step 5.3.2.6.3.2

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

$(3) (9 (3) - 8) \geq 0$

Step 5.3.2.6.3.3

The left side $57$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 5.3.2.6.4

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

$x < 0$ True

$0 < x < \frac{8}{9}$ False

$x > \frac{8}{9}$ True

$x < 0$ True

$0 < x < \frac{8}{9}$ False

$x > \frac{8}{9}$ True

Step 5.3.2.7

The solution consists of all of the true intervals.

$x \leq 0$ or $x \geq \frac{8}{9}$

Step 5.3.3

Set the denominator in $\frac{x - 2 - \sqrt{x (9 x - 8)}}{2 (x - 1)}$ equal to $0$ to find where the expression is undefined.

$2 (x - 1) = 0$

Step 5.3.4

Solve for $x$ .

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Step 5.3.4.1

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

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Step 5.3.4.1.1

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

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

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.

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

Step 5.3.4.1.2.1.2

Divide $x - 1$ by $1$ .

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

Step 5.3.4.1.3

Simplify the right side.

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Step 5.3.4.1.3.1

Divide $0$ by $2$ .

$x - 1 = 0$

Step 5.3.4.2

Add $1$ to both sides of the equation.

$x = 1$

Step 5.3.5

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

$(- \infty, 0] \cup [\frac{8}{9}, 1) \cup (1, \infty)$

Step 5.4

Since the domain of $f^{- 1} (x) = - \frac{x - 2 - \sqrt{x (9 x - 8)}}{2 (x - 1)}, - \frac{x - 2 + \sqrt{x (9 x - 8)}}{2 (x - 1)}$ is not equal to the range of $f (x) = \frac{x^{2} + 2 x + 1}{x^{2} + x - 2}$ , then $f^{- 1} (x) = - \frac{x - 2 - \sqrt{x (9 x - 8)}}{2 (x - 1)}, - \frac{x - 2 + \sqrt{x (9 x - 8)}}{2 (x - 1)}$ is not an inverse of $f (x) = \frac{x^{2} + 2 x + 1}{x^{2} + x - 2}$ .

There is no inverse

Step 6