Precalculus Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

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.

$y, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$y$

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $y$ .

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Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

Rewrite the expression.

$y - 1 = x y$

Step 3.4

Solve the equation.

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Step 3.4.1

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

$y - 1 - x y = 0$

Step 3.4.2

Add $1$ to both sides of the equation.

$y - x y = 1$

Step 3.4.3

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

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Step 3.4.3.1

Factor $y$ out of $y^{1}$ .

$y \cdot 1 - x y = 1$

Step 3.4.3.2

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

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

Step 3.4.3.3

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

$y (1 - x) = 1$

Step 3.4.4

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

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Step 3.4.4.1

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

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

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

Cancel the common factor of $1 - x$ .

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Step 3.4.4.2.1.1

Cancel the common factor.

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

Step 3.4.4.2.1.2

Divide $y$ by $1$ .

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Simplify the denominator.

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Step 5.2.3.1

Write $1$ as a fraction with a common denominator.

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

Step 5.2.3.2

Combine the numerators over the common denominator.

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

Step 5.2.3.3

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

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Step 5.2.3.3.1

Apply the distributive property.

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

Step 5.2.3.3.2

Multiply $- 1$ by $- 1$ .

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

Step 5.2.3.3.3

Subtract $x$ from $x$ .

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

Step 5.2.3.3.4

Add $0$ and $1$ .

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

Step 5.2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2.5

Multiply $x$ by $1$ .

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

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

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

Step 5.3.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.4

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

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

Step 5.3.5

Simplify terms.

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Step 5.3.5.1

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

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

Step 5.3.5.2

Combine the numerators over the common denominator.

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

Step 5.3.6

Simplify the numerator.

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Step 5.3.6.1

Apply the distributive property.

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

Step 5.3.6.2

Multiply $- 1$ by $1$ .

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

Step 5.3.6.3

Multiply $- - x$ .

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Step 5.3.6.3.1

Multiply $- 1$ by $- 1$ .

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

Step 5.3.6.3.2

Multiply $x$ by $1$ .

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

Step 5.3.6.4

Subtract $1$ from $1$ .

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

Step 5.3.6.5

Add $0$ and $x$ .

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

Step 5.3.7

Cancel the common factor of $1 - x$ .

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Step 5.3.7.1

Cancel the common factor.

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

Step 5.3.7.2

Rewrite the expression.

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

Step 5.4

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

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