Precalculus Examples

$f (x) = \frac{3}{x - 1} + 2$

Step 1

Write $f (x) = \frac{3}{x - 1} + 2$ as an equation.

$y = \frac{3}{x - 1} + 2$

Step 2

Interchange the variables.

$x = \frac{3}{y - 1} + 2$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{3}{y - 1} + 2 = x$ .

$\frac{3}{y - 1} + 2 = x$

Step 3.2

Subtract $2$ from both sides of the equation.

$\frac{3}{y - 1} = x - 2$

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.

$y - 1, 1, 1$

Step 3.3.2

Remove parentheses.

$y - 1, 1, 1$

Step 3.3.3

The LCM of one and any expression is the expression.

$y - 1$

Step 3.4

Multiply each term in $\frac{3}{y - 1} = x - 2$ by $y - 1$ to eliminate the fractions.

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Step 3.4.1

Multiply each term in $\frac{3}{y - 1} = x - 2$ by $y - 1$ .

$\frac{3}{y - 1} (y - 1) = x (y - 1) - 2 (y - 1)$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $y - 1$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{3}{y - 1} (y - 1) = x (y - 1) - 2 (y - 1)$

Step 3.4.2.1.2

Rewrite the expression.

$3 = x (y - 1) - 2 (y - 1)$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify each term.

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Step 3.4.3.1.1

Apply the distributive property.

$3 = x y + x \cdot - 1 - 2 (y - 1)$

Step 3.4.3.1.2

Move $- 1$ to the left of $x$ .

$3 = x y - 1 \cdot x - 2 (y - 1)$

Step 3.4.3.1.3

Rewrite $- 1 x$ as $- x$ .

$3 = x y - x - 2 (y - 1)$

Step 3.4.3.1.4

Apply the distributive property.

$3 = x y - x - 2 y - 2 \cdot - 1$

Step 3.4.3.1.5

Multiply $- 2$ by $- 1$ .

$3 = x y - x - 2 y + 2$

Step 3.5

Solve the equation.

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Step 3.5.1

Rewrite the equation as $x y - x - 2 y + 2 = 3$ .

$x y - x - 2 y + 2 = 3$

Step 3.5.2

Move all terms not containing $y$ to the right side of the equation.

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Step 3.5.2.1

Add $x$ to both sides of the equation.

$x y - 2 y + 2 = 3 + x$

Step 3.5.2.2

Subtract $2$ from both sides of the equation.

$x y - 2 y = 3 + x - 2$

Step 3.5.2.3

Subtract $2$ from $3$ .

$x y - 2 y = x + 1$

Step 3.5.3

Factor $y$ out of $x y - 2 y$ .

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Step 3.5.3.1

Factor $y$ out of $x y$ .

$y x - 2 y = x + 1$

Step 3.5.3.2

Factor $y$ out of $- 2 y$ .

$y x + y \cdot - 2 = x + 1$

Step 3.5.3.3

Factor $y$ out of $y x + y \cdot - 2$ .

$y (x - 2) = x + 1$

Step 3.5.4

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

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Step 3.5.4.1

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

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

Step 3.5.4.2

Simplify the left side.

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Step 3.5.4.2.1

Cancel the common factor of $x - 2$ .

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Step 3.5.4.2.1.1

Cancel the common factor.

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

Step 3.5.4.2.1.2

Divide $y$ by $1$ .

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

Step 3.5.4.3

Simplify the right side.

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Step 3.5.4.3.1

Combine the numerators over the common denominator.

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

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = \frac{x + 1}{x - 2}$ is the inverse of $f (x) = \frac{3}{x - 1} + 2$ .

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Step 5.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 5.2

Evaluate $f^{- 1} (f (x))$ .

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Step 5.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 5.2.2

Evaluate $f^{- 1} (\frac{3}{x - 1} + 2)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{3}{x - 1} + 2) = \frac{(\frac{3}{x - 1} + 2) + 1}{(\frac{3}{x - 1} + 2) - 2}$

Step 5.2.3

Multiply the numerator and denominator of the fraction by $x - 1$ .

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Step 5.2.3.1

Multiply $\frac{\frac{3}{x - 1} + 2 + 1}{\frac{3}{x - 1} + 2 - 2}$ by $\frac{x - 1}{x - 1}$ .

$f^{- 1} (\frac{3}{x - 1} + 2) = \frac{x - 1}{x - 1} \cdot \frac{\frac{3}{x - 1} + 2 + 1}{\frac{3}{x - 1} + 2 - 2}$

Step 5.2.3.2

Combine.

$f^{- 1} (\frac{3}{x - 1} + 2) = \frac{(x - 1) (\frac{3}{x - 1} + 2 + 1)}{(x - 1) (\frac{3}{x - 1} + 2 - 2)}$

Step 5.2.4

Apply the distributive property.

$f^{- 1} (\frac{3}{x - 1} + 2) = \frac{(x - 1) (\frac{3}{x - 1}) + (x - 1) \cdot 2 + (x - 1) \cdot 1}{(x - 1) (\frac{3}{x - 1}) + (x - 1) \cdot 2 + (x - 1) \cdot - 2}$

Step 5.2.5

Simplify by cancelling.

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Step 5.2.5.1

Cancel the common factor of $x - 1$ .

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Step 5.2.5.1.1

Cancel the common factor.

$f^{- 1} (\frac{3}{x - 1} + 2) = \frac{(x - 1) (\frac{3}{x - 1}) + (x - 1) \cdot 2 + (x - 1) \cdot 1}{(x - 1) (\frac{3}{x - 1}) + (x - 1) \cdot 2 + (x - 1) \cdot - 2}$

Step 5.2.5.1.2

Rewrite the expression.

$f^{- 1} (\frac{3}{x - 1} + 2) = \frac{3 + (x - 1) \cdot 2 + (x - 1) \cdot 1}{(x - 1) (\frac{3}{x - 1}) + (x - 1) \cdot 2 + (x - 1) \cdot - 2}$

Step 5.2.5.2

Cancel the common factor of $x - 1$ .

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Step 5.2.5.2.1

Cancel the common factor.

$f^{- 1} (\frac{3}{x - 1} + 2) = \frac{3 + (x - 1) \cdot 2 + (x - 1) \cdot 1}{(x - 1) (\frac{3}{x - 1}) + (x - 1) \cdot 2 + (x - 1) \cdot - 2}$

Step 5.2.5.2.2

Rewrite the expression.

$f^{- 1} (\frac{3}{x - 1} + 2) = \frac{3 + (x - 1) \cdot 2 + (x - 1) \cdot 1}{3 + (x - 1) \cdot 2 + (x - 1) \cdot - 2}$

Step 5.2.6

Simplify the numerator.

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Step 5.2.6.1

Reorder $x - 1$ and $1$ .

$f^{- 1} (\frac{3}{x - 1} + 2) = \frac{3 + 2 \cdot (x - 1) + 1 \cdot (x - 1)}{3 + (x - 1) \cdot 2 + (x - 1) \cdot - 2}$

Step 5.2.6.2

Add $2 \cdot (x - 1)$ and $1 \cdot (x - 1)$ .

$f^{- 1} (\frac{3}{x - 1} + 2) = \frac{3 + 3 \cdot (x - 1)}{3 + (x - 1) \cdot 2 + (x - 1) \cdot - 2}$

Step 5.2.6.3

Factor $3$ out of $3 + 3 \cdot (x - 1)$ .

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Step 5.2.6.3.1

Factor $3$ out of $3$ .

$f^{- 1} (\frac{3}{x - 1} + 2) = \frac{3 \cdot 1 + 3 \cdot (x - 1)}{3 + (x - 1) \cdot 2 + (x - 1) \cdot - 2}$

Step 5.2.6.3.2

Factor $3$ out of $3 \cdot 1 + 3 \cdot (x - 1)$ .

$f^{- 1} (\frac{3}{x - 1} + 2) = \frac{3 (1 + x - 1)}{3 + (x - 1) \cdot 2 + (x - 1) \cdot - 2}$

Step 5.2.6.4

Subtract $1$ from $1$ .

$f^{- 1} (\frac{3}{x - 1} + 2) = \frac{3 (x + 0)}{3 + (x - 1) \cdot 2 + (x - 1) \cdot - 2}$

Step 5.2.6.5

Add $x$ and $0$ .

$f^{- 1} (\frac{3}{x - 1} + 2) = \frac{3 x}{3 + (x - 1) \cdot 2 + (x - 1) \cdot - 2}$

Step 5.2.7

Simplify the denominator.

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Step 5.2.7.1

Apply the distributive property.

$f^{- 1} (\frac{3}{x - 1} + 2) = \frac{3 x}{3 + x \cdot 2 - 1 \cdot 2 + (x - 1) \cdot - 2}$

Step 5.2.7.2

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

$f^{- 1} (\frac{3}{x - 1} + 2) = \frac{3 x}{3 + 2 \cdot x - 1 \cdot 2 + (x - 1) \cdot - 2}$

Step 5.2.7.3

Multiply $- 1$ by $2$ .

$f^{- 1} (\frac{3}{x - 1} + 2) = \frac{3 x}{3 + 2 \cdot x - 2 + (x - 1) \cdot - 2}$

Step 5.2.7.4

Apply the distributive property.

$f^{- 1} (\frac{3}{x - 1} + 2) = \frac{3 x}{3 + 2 x - 2 + x \cdot - 2 - 1 \cdot - 2}$

Step 5.2.7.5

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

$f^{- 1} (\frac{3}{x - 1} + 2) = \frac{3 x}{3 + 2 x - 2 - 2 \cdot x - 1 \cdot - 2}$

Step 5.2.7.6

Multiply $- 1$ by $- 2$ .

$f^{- 1} (\frac{3}{x - 1} + 2) = \frac{3 x}{3 + 2 x - 2 - 2 \cdot x + 2}$

Step 5.2.7.7

Subtract $2$ from $3$ .

$f^{- 1} (\frac{3}{x - 1} + 2) = \frac{3 x}{2 x + 1 - 2 x + 2}$

Step 5.2.7.8

Subtract $2 x$ from $2 x$ .

$f^{- 1} (\frac{3}{x - 1} + 2) = \frac{3 x}{0 + 1 + 2}$

Step 5.2.7.9

Add $0$ and $1$ .

$f^{- 1} (\frac{3}{x - 1} + 2) = \frac{3 x}{1 + 2}$

Step 5.2.7.10

Add $1$ and $2$ .

$f^{- 1} (\frac{3}{x - 1} + 2) = \frac{3 x}{3}$

Step 5.2.8

Cancel the common factor of $3$ .

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Step 5.2.8.1

Cancel the common factor.

$f^{- 1} (\frac{3}{x - 1} + 2) = \frac{3 x}{3}$

Step 5.2.8.2

Divide $x$ by $1$ .

$f^{- 1} (\frac{3}{x - 1} + 2) = x$

Step 5.3

Evaluate $f (f^{- 1} (x))$ .

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Step 5.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 5.3.2

Evaluate $f (\frac{x + 1}{x - 2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x + 1}{x - 2}) = \frac{3}{(\frac{x + 1}{x - 2}) - 1} + 2$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Simplify the denominator.

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Step 5.3.3.1.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$f (\frac{x + 1}{x - 2}) = \frac{3}{\frac{x + 1}{x - 2} - 1 \cdot \frac{x - 2}{x - 2}} + 2$

Step 5.3.3.1.2

Combine $- 1$ and $\frac{x - 2}{x - 2}$ .

$f (\frac{x + 1}{x - 2}) = \frac{3}{\frac{x + 1}{x - 2} + \frac{- (x - 2)}{x - 2}} + 2$

Step 5.3.3.1.3

Combine the numerators over the common denominator.

$f (\frac{x + 1}{x - 2}) = \frac{3}{\frac{x + 1 - (x - 2)}{x - 2}} + 2$

Step 5.3.3.1.4

Rewrite $\frac{x + 1 - (x - 2)}{x - 2}$ in a factored form.

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Step 5.3.3.1.4.1

Apply the distributive property.

$f (\frac{x + 1}{x - 2}) = \frac{3}{\frac{x + 1 - x + 2}{x - 2}} + 2$

Step 5.3.3.1.4.2

Multiply $- 1$ by $- 2$ .

$f (\frac{x + 1}{x - 2}) = \frac{3}{\frac{x + 1 - x + 2}{x - 2}} + 2$

Step 5.3.3.1.4.3

Subtract $x$ from $x$ .

$f (\frac{x + 1}{x - 2}) = \frac{3}{\frac{0 + 1 + 2}{x - 2}} + 2$

Step 5.3.3.1.4.4

Add $0$ and $1$ .

$f (\frac{x + 1}{x - 2}) = \frac{3}{\frac{1 + 2}{x - 2}} + 2$

Step 5.3.3.1.4.5

Add $1$ and $2$ .

$f (\frac{x + 1}{x - 2}) = \frac{3}{\frac{3}{x - 2}} + 2$

Step 5.3.3.2

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{x + 1}{x - 2}) = 3 (\frac{x - 2}{3}) + 2$

Step 5.3.3.3

Cancel the common factor of $3$ .

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Step 5.3.3.3.1

Cancel the common factor.

$f (\frac{x + 1}{x - 2}) = 3 (\frac{x - 2}{3}) + 2$

Step 5.3.3.3.2

Rewrite the expression.

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

Step 5.3.4

Combine the opposite terms in $x - 2 + 2$ .

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Step 5.3.4.1

Add $- 2$ and $2$ .

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

Step 5.3.4.2

Add $x$ and $0$ .

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

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x + 1}{x - 2}$ is the inverse of $f (x) = \frac{3}{x - 1} + 2$ .

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