Finite Math Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

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.

$y - 2, 1$

Step 3.2.2

Remove parentheses.

$y - 2, 1$

Step 3.2.3

The LCM of one and any expression is the expression.

$y - 2$

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $y - 2$ .

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Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

Rewrite the expression.

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

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Apply the distributive property.

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

Step 3.3.3.2

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

$2 y - 1 = x y - 2 x$

Step 3.4

Solve the equation.

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Step 3.4.1

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

$2 y - 1 - x y = - 2 x$

Step 3.4.2

Add $1$ to both sides of the equation.

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

Step 3.4.3

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

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Step 3.4.3.1

Factor $y$ out of $2 y$ .

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

Step 3.4.3.2

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

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

Step 3.4.3.3

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

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

Step 3.4.4

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

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Step 3.4.4.1

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

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

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

Cancel the common factor of $2 - x$ .

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Step 3.4.4.2.1.1

Cancel the common factor.

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

Step 3.4.4.2.1.2

Divide $y$ by $1$ .

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

Step 3.4.4.3

Simplify the right side.

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Step 3.4.4.3.1

Combine the numerators over the common denominator.

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

Step 3.4.4.3.2

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

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

Step 3.4.4.3.3

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

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

Step 3.4.4.3.4

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

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

Step 3.4.4.3.5

Simplify the expression.

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Step 3.4.4.3.5.1

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

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

Step 3.4.4.3.5.2

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Multiply the numerator and denominator of the fraction by $x - 2$ .

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Step 5.2.3.1

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

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

Step 5.2.3.2

Combine.

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

Step 5.2.4

Apply the distributive property.

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

Step 5.2.5

Cancel the common factor of $x - 2$ .

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Step 5.2.5.1

Move the leading negative in $- \frac{2 x - 1}{x - 2}$ into the numerator.

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

Step 5.2.5.2

Cancel the common factor.

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

Step 5.2.5.3

Rewrite the expression.

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

Step 5.2.6

Simplify the numerator.

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Step 5.2.6.1

Factor $x - 2$ out of $(x - 2) \cdot 2 \frac{2 x - 1}{x - 2} + (x - 2) \cdot - 1$ .

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Step 5.2.6.1.1

Factor $x - 2$ out of $(x - 2) \cdot 2 \frac{2 x - 1}{x - 2}$ .

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

Step 5.2.6.1.2

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

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

Step 5.2.6.1.3

Factor $x - 2$ out of $(x - 2) (2 \frac{2 x - 1}{x - 2}) + (x - 2) (- 1)$ .

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

Step 5.2.6.2

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

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

Step 5.2.6.3

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

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

Step 5.2.6.4

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

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

Step 5.2.6.5

Combine the numerators over the common denominator.

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

Step 5.2.6.6

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

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Step 5.2.6.6.1

Apply the distributive property.

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

Step 5.2.6.6.2

Multiply $2$ by $2$ .

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

Step 5.2.6.6.3

Multiply $2$ by $- 1$ .

$f^{- 1} (\frac{2 x - 1}{x - 2}) = - \frac{(x - 2) (\frac{4 x - 2 - (x - 2)}{x - 2})}{(x - 2) \cdot 2 - (2 x - 1)}$

Step 5.2.6.6.4

Apply the distributive property.

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

Step 5.2.6.6.5

Multiply $- 1$ by $- 2$ .

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

Step 5.2.6.6.6

Subtract $x$ from $4 x$ .

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

Step 5.2.6.6.7

Add $- 2$ and $2$ .

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

Step 5.2.6.6.8

Add $3 x$ and $0$ .

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

Step 5.2.7

Simplify the denominator.

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Step 5.2.7.1

Apply the distributive property.

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

Step 5.2.7.2

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

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

Step 5.2.7.3

Multiply $- 2$ by $2$ .

$f^{- 1} (\frac{2 x - 1}{x - 2}) = - \frac{(x - 2) (\frac{3 x}{x - 2})}{2 \cdot x - 4 - (2 x - 1)}$

Step 5.2.7.4

Apply the distributive property.

$f^{- 1} (\frac{2 x - 1}{x - 2}) = - \frac{(x - 2) (\frac{3 x}{x - 2})}{2 x - 4 - (2 x) + 1}$

Step 5.2.7.5

Multiply $2$ by $- 1$ .

$f^{- 1} (\frac{2 x - 1}{x - 2}) = - \frac{(x - 2) (\frac{3 x}{x - 2})}{2 x - 4 - 2 x + 1}$

Step 5.2.7.6

Multiply $- 1$ by $- 1$ .

$f^{- 1} (\frac{2 x - 1}{x - 2}) = - \frac{(x - 2) (\frac{3 x}{x - 2})}{2 x - 4 - 2 x + 1}$

Step 5.2.7.7

Subtract $2 x$ from $2 x$ .

$f^{- 1} (\frac{2 x - 1}{x - 2}) = - \frac{(x - 2) (\frac{3 x}{x - 2})}{0 - 4 + 1}$

Step 5.2.7.8

Subtract $4$ from $0$ .

$f^{- 1} (\frac{2 x - 1}{x - 2}) = - \frac{(x - 2) (\frac{3 x}{x - 2})}{- 4 + 1}$

Step 5.2.7.9

Add $- 4$ and $1$ .

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

Step 5.2.8

Simplify terms.

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Step 5.2.8.1

Factor $\frac{3 x}{x - 2}$ out of $\frac{(x - 2) \frac{3 x}{x - 2}}{- 3}$ .

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

Step 5.2.8.2

Cancel the common factor of $3$ .

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Step 5.2.8.2.1

Factor $3$ out of $3 x$ .

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

Step 5.2.8.2.2

Factor $3$ out of $- 3$ .

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

Step 5.2.8.2.3

Cancel the common factor.

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

Step 5.2.8.2.4

Rewrite the expression.

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

Step 5.2.8.3

Cancel the common factor of $x - 2$ .

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Step 5.2.8.3.1

Cancel the common factor.

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

Step 5.2.8.3.2

Rewrite the expression.

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

Step 5.2.8.4

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

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

Step 5.2.8.5

Simplify the expression.

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Step 5.2.8.5.1

Move the negative one from the denominator of $\frac{x}{- 1}$ .

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

Step 5.2.8.5.2

Rewrite $- 1 \cdot x$ as $- x$ .

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

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

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

Step 5.3.3

Multiply the numerator and denominator of the fraction by $2 - x$ .

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Step 5.3.3.1

Multiply $\frac{2 (- \frac{2 x - 1}{2 - x}) - 1}{- \frac{2 x - 1}{2 - x} - 2}$ by $\frac{2 - x}{2 - x}$ .

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

Step 5.3.3.2

Combine.

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

Step 5.3.4

Apply the distributive property.

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

Step 5.3.5

Cancel the common factor of $2 - x$ .

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Step 5.3.5.1

Move the leading negative in $- \frac{2 x - 1}{2 - x}$ into the numerator.

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

Step 5.3.5.2

Cancel the common factor.

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

Step 5.3.5.3

Rewrite the expression.

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

Step 5.3.6

Simplify the numerator.

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Step 5.3.6.1

Factor $2 - x$ out of $(2 - x) \cdot 2 (- \frac{2 x - 1}{2 - x}) + (2 - x) \cdot - 1$ .

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Step 5.3.6.1.1

Factor $2 - x$ out of $(2 - x) \cdot 2 (- \frac{2 x - 1}{2 - x})$ .

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

Step 5.3.6.1.2

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

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

Step 5.3.6.1.3

Factor $2 - x$ out of $(2 - x) (2 (- \frac{2 x - 1}{2 - x})) + (2 - x) (- 1)$ .

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

Step 5.3.6.2

Multiply $2 (- \frac{2 x - 1}{2 - x})$ .

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Step 5.3.6.2.1

Multiply $- 1$ by $2$ .

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

Step 5.3.6.2.2

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

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

Step 5.3.6.3

Move the negative in front of the fraction.

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

Step 5.3.6.4

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

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

Step 5.3.6.5

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

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

Step 5.3.6.6

Combine the numerators over the common denominator.

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

Step 5.3.6.7

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

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Step 5.3.6.7.1

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

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Step 5.3.6.7.1.1

Reorder $2$ and $- x$ .

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

Step 5.3.6.7.1.2

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

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

Step 5.3.6.7.1.3

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

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

Step 5.3.6.7.2

Apply the distributive property.

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

Step 5.3.6.7.3

Multiply $2$ by $2$ .

$f (- \frac{2 x - 1}{2 - x}) = \frac{(2 - x) (\frac{- (4 x + 2 \cdot - 1 - x + 2)}{2 - x})}{- (2 x - 1) + (2 - x) \cdot - 2}$

Step 5.3.6.7.4

Multiply $2$ by $- 1$ .

$f (- \frac{2 x - 1}{2 - x}) = \frac{(2 - x) (\frac{- (4 x - 2 - x + 2)}{2 - x})}{- (2 x - 1) + (2 - x) \cdot - 2}$

Step 5.3.6.7.5

Subtract $x$ from $4 x$ .

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

Step 5.3.6.7.6

Add $- 2$ and $2$ .

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

Step 5.3.6.7.7

Add $3 x$ and $0$ .

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

Step 5.3.6.7.8

Multiply $- 1$ by $3$ .

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

Step 5.3.6.8

Move the negative in front of the fraction.

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

Step 5.3.6.9

Remove unnecessary parentheses.

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

Step 5.3.6.10

Factor out negative.

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

Step 5.3.7

Simplify the denominator.

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Step 5.3.7.1

Apply the distributive property.

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

Step 5.3.7.2

Multiply $2$ by $- 1$ .

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

Step 5.3.7.3

Multiply $- 1$ by $- 1$ .

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

Step 5.3.7.4

Apply the distributive property.

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

Step 5.3.7.5

Multiply $2$ by $- 2$ .

$f (- \frac{2 x - 1}{2 - x}) = \frac{- (2 - x) \frac{3 x}{2 - x}}{- 2 x + 1 - 4 - x \cdot - 2}$

Step 5.3.7.6

Multiply $- 2$ by $- 1$ .

$f (- \frac{2 x - 1}{2 - x}) = \frac{- (2 - x) \frac{3 x}{2 - x}}{- 2 x + 1 - 4 + 2 x}$

Step 5.3.7.7

Add $- 2 x$ and $2 x$ .

$f (- \frac{2 x - 1}{2 - x}) = \frac{- (2 - x) \frac{3 x}{2 - x}}{0 + 1 - 4}$

Step 5.3.7.8

Add $0$ and $1$ .

$f (- \frac{2 x - 1}{2 - x}) = \frac{- (2 - x) \frac{3 x}{2 - x}}{1 - 4}$

Step 5.3.7.9

Subtract $4$ from $1$ .

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

Step 5.3.8

Simplify terms.

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Step 5.3.8.1

Factor $\frac{3 x}{2 - x}$ out of $\frac{- (2 - x) \frac{3 x}{2 - x}}{- 3}$ .

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

Step 5.3.8.2

Cancel the common factor of $3$ .

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Step 5.3.8.2.1

Factor $3$ out of $3 x$ .

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

Step 5.3.8.2.2

Factor $3$ out of $- 3$ .

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

Step 5.3.8.2.3

Cancel the common factor.

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

Step 5.3.8.2.4

Rewrite the expression.

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

Step 5.3.8.3

Cancel the common factor of $2 - x$ .

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Step 5.3.8.3.1

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

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

Step 5.3.8.3.2

Cancel the common factor.

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

Step 5.3.8.3.3

Rewrite the expression.

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

Step 5.3.8.4

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

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

Step 5.3.8.5

Simplify the expression.

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Step 5.3.8.5.1

Move the negative one from the denominator of $\frac{x \cdot - 1}{- 1}$ .

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

Step 5.3.8.5.2

Rewrite $- 1 \cdot (x \cdot - 1)$ as $- (x \cdot - 1)$ .

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

Step 5.3.8.5.3

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

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

Step 5.3.8.5.4

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

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

Step 5.4

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

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