Algebra Examples

$f (x) = \frac{x + 20}{x - 5}$

Step 1

Write $f (x) = \frac{x + 20}{x - 5}$ as an equation.

$y = \frac{x + 20}{x - 5}$

Step 2

Interchange the variables.

$x = \frac{y + 20}{y - 5}$

Step 3

Solve for $y$ .

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Step 3.1

Multiply the equation by $y - 5$ .

$(y - 5) (x) = (\frac{y + 20}{y - 5}) (y - 5)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Apply the distributive property.

$y x - 5 x = (\frac{y + 20}{y - 5}) (y - 5)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $y - 5$ .

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Step 3.3.1.1

Cancel the common factor.

$y x - 5 x = \frac{y + 20}{y - 5} (y - 5)$

Step 3.3.1.2

Rewrite the expression.

$y x - 5 x = y + 20$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $y$ from both sides of the equation.

$y x - 5 x - y = 20$

Step 3.4.2

Add $5 x$ to both sides of the equation.

$y x - y = 20 + 5 x$

Step 3.4.3

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

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Step 3.4.3.1

Factor $y$ out of $y x$ .

$y (x) - y = 20 + 5 x$

Step 3.4.3.2

Factor $y$ out of $- y$ .

$y (x) + y \cdot - 1 = 20 + 5 x$

Step 3.4.3.3

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

$y (x - 1) = 20 + 5 x$

Step 3.4.4

Divide each term in $y (x - 1) = 20 + 5 x$ by $x - 1$ and simplify.

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Step 3.4.4.1

Divide each term in $y (x - 1) = 20 + 5 x$ by $x - 1$ .

$\frac{y (x - 1)}{x - 1} = \frac{20}{x - 1} + \frac{5 x}{x - 1}$

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

Cancel the common factor of $x - 1$ .

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Step 3.4.4.2.1.1

Cancel the common factor.

$\frac{y (x - 1)}{x - 1} = \frac{20}{x - 1} + \frac{5 x}{x - 1}$

Step 3.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{20}{x - 1} + \frac{5 x}{x - 1}$

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.

$y = \frac{20 + 5 x}{x - 1}$

Step 3.4.4.3.2

Factor $5$ out of $20 + 5 x$ .

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Step 3.4.4.3.2.1

Factor $5$ out of $20$ .

$y = \frac{5 \cdot 4 + 5 x}{x - 1}$

Step 3.4.4.3.2.2

Factor $5$ out of $5 \cdot 4 + 5 x$ .

$y = \frac{5 (4 + x)}{x - 1}$

Step 4

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

$f^{- 1} (x) = \frac{5 (4 + x)}{x - 1}$

Step 5

Verify if $f^{- 1} (x) = \frac{5 (4 + x)}{x - 1}$ is the inverse of $f (x) = \frac{x + 20}{x - 5}$ .

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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 $f^{- 1} (\frac{x + 20}{x - 5})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x + 20}{x - 5}) = \frac{5 (4 + \frac{x + 20}{x - 5})}{(\frac{x + 20}{x - 5}) - 1}$

Step 5.2.3

Remove parentheses.

$f^{- 1} (\frac{x + 20}{x - 5}) = \frac{5 (4 + \frac{x + 20}{x - 5})}{(\frac{x + 20}{x - 5}) - 1}$

Step 5.2.4

Simplify the numerator.

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Step 5.2.4.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$f^{- 1} (\frac{x + 20}{x - 5}) = \frac{5 (\frac{4 (x - 5)}{x - 5} + \frac{x + 20}{x - 5})}{\frac{x + 20}{x - 5} - 1}$

Step 5.2.4.2

Combine the numerators over the common denominator.

$f^{- 1} (\frac{x + 20}{x - 5}) = \frac{5 (\frac{4 (x - 5) + x + 20}{x - 5})}{\frac{x + 20}{x - 5} - 1}$

Step 5.2.4.3

Reorder terms.

$f^{- 1} (\frac{x + 20}{x - 5}) = \frac{5 (\frac{x + 4 (x - 5) + 20}{x - 5})}{\frac{x + 20}{x - 5} - 1}$

Step 5.2.4.4

Rewrite $\frac{x + 4 (x - 5) + 20}{x - 5}$ in a factored form.

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Step 5.2.4.4.1

Apply the distributive property.

$f^{- 1} (\frac{x + 20}{x - 5}) = \frac{5 (\frac{x + 4 x + 4 \cdot - 5 + 20}{x - 5})}{\frac{x + 20}{x - 5} - 1}$

Step 5.2.4.4.2

Multiply $4$ by $- 5$ .

$f^{- 1} (\frac{x + 20}{x - 5}) = \frac{5 (\frac{x + 4 x - 20 + 20}{x - 5})}{\frac{x + 20}{x - 5} - 1}$

Step 5.2.4.4.3

Add $x$ and $4 x$ .

$f^{- 1} (\frac{x + 20}{x - 5}) = \frac{5 (\frac{5 x - 20 + 20}{x - 5})}{\frac{x + 20}{x - 5} - 1}$

Step 5.2.4.4.4

Add $- 20$ and $20$ .

$f^{- 1} (\frac{x + 20}{x - 5}) = \frac{5 (\frac{5 x + 0}{x - 5})}{\frac{x + 20}{x - 5} - 1}$

Step 5.2.4.4.5

Add $5 x$ and $0$ .

$f^{- 1} (\frac{x + 20}{x - 5}) = \frac{5 (\frac{5 x}{x - 5})}{\frac{x + 20}{x - 5} - 1}$

Step 5.2.5

Simplify the denominator.

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Step 5.2.5.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$f^{- 1} (\frac{x + 20}{x - 5}) = \frac{5 (\frac{5 x}{x - 5})}{\frac{x + 20}{x - 5} - 1 \cdot \frac{x - 5}{x - 5}}$

Step 5.2.5.2

Combine $- 1$ and $\frac{x - 5}{x - 5}$ .

$f^{- 1} (\frac{x + 20}{x - 5}) = \frac{5 (\frac{5 x}{x - 5})}{\frac{x + 20}{x - 5} + \frac{- (x - 5)}{x - 5}}$

Step 5.2.5.3

Combine the numerators over the common denominator.

$f^{- 1} (\frac{x + 20}{x - 5}) = \frac{5 (\frac{5 x}{x - 5})}{\frac{x + 20 - (x - 5)}{x - 5}}$

Step 5.2.5.4

Rewrite $\frac{x + 20 - (x - 5)}{x - 5}$ in a factored form.

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Step 5.2.5.4.1

Apply the distributive property.

$f^{- 1} (\frac{x + 20}{x - 5}) = \frac{5 (\frac{5 x}{x - 5})}{\frac{x + 20 - x + 5}{x - 5}}$

Step 5.2.5.4.2

Multiply $- 1$ by $- 5$ .

$f^{- 1} (\frac{x + 20}{x - 5}) = \frac{5 (\frac{5 x}{x - 5})}{\frac{x + 20 - x + 5}{x - 5}}$

Step 5.2.5.4.3

Subtract $x$ from $x$ .

$f^{- 1} (\frac{x + 20}{x - 5}) = \frac{5 (\frac{5 x}{x - 5})}{\frac{0 + 20 + 5}{x - 5}}$

Step 5.2.5.4.4

Add $0$ and $20$ .

$f^{- 1} (\frac{x + 20}{x - 5}) = \frac{5 (\frac{5 x}{x - 5})}{\frac{20 + 5}{x - 5}}$

Step 5.2.5.4.5

Add $20$ and $5$ .

$f^{- 1} (\frac{x + 20}{x - 5}) = \frac{5 (\frac{5 x}{x - 5})}{\frac{25}{x - 5}}$

Step 5.2.6

Combine $5$ and $\frac{5 x}{x - 5}$ .

$f^{- 1} (\frac{x + 20}{x - 5}) = \frac{\frac{5 (5 x)}{x - 5}}{\frac{25}{x - 5}}$

Step 5.2.7

Multiply $5$ by $5$ .

$f^{- 1} (\frac{x + 20}{x - 5}) = \frac{\frac{25 x}{x - 5}}{\frac{25}{x - 5}}$

Step 5.2.8

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{x + 20}{x - 5}) = \frac{25 x}{x - 5} \cdot \frac{x - 5}{25}$

Step 5.2.9

Cancel the common factor of $25$ .

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Step 5.2.9.1

Factor $25$ out of $25 x$ .

$f^{- 1} (\frac{x + 20}{x - 5}) = \frac{25 (x)}{x - 5} \cdot \frac{x - 5}{25}$

Step 5.2.9.2

Cancel the common factor.

$f^{- 1} (\frac{x + 20}{x - 5}) = \frac{25 x}{x - 5} \cdot \frac{x - 5}{25}$

Step 5.2.9.3

Rewrite the expression.

$f^{- 1} (\frac{x + 20}{x - 5}) = \frac{x}{x - 5} \cdot (x - 5)$

Step 5.2.10

Cancel the common factor of $x - 5$ .

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Step 5.2.10.1

Cancel the common factor.

$f^{- 1} (\frac{x + 20}{x - 5}) = \frac{x}{x - 5} \cdot (x - 5)$

Step 5.2.10.2

Rewrite the expression.

$f^{- 1} (\frac{x + 20}{x - 5}) = x$

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 $f (\frac{5 (4 + x)}{x - 1})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{5 (4 + x)}{x - 1}) = \frac{(\frac{5 (4 + x)}{x - 1}) + 20}{(\frac{5 (4 + x)}{x - 1}) - 5}$

Step 5.3.3

Cancel the common factor of $(\frac{5 (4 + x)}{x - 1}) + 20$ and $(\frac{5 (4 + x)}{x - 1}) - 5$ .

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Step 5.3.3.1

Factor $5$ out of $\frac{5 (4 + x)}{x - 1}$ .

$f (\frac{5 (4 + x)}{x - 1}) = \frac{5 (\frac{4 + x}{x - 1}) + 20}{(\frac{5 (4 + x)}{x - 1}) - 5}$

Step 5.3.3.2

Factor $5$ out of $20$ .

$f (\frac{5 (4 + x)}{x - 1}) = \frac{5 (\frac{4 + x}{x - 1}) + 5 (4)}{(\frac{5 (4 + x)}{x - 1}) - 5}$

Step 5.3.3.3

Factor $5$ out of $5 \frac{4 + x}{x - 1} + 5 (4)$ .

$f (\frac{5 (4 + x)}{x - 1}) = \frac{5 (\frac{4 + x}{x - 1} + 4)}{(\frac{5 (4 + x)}{x - 1}) - 5}$

Step 5.3.3.4

Cancel the common factors.

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Step 5.3.3.4.1

Factor $5$ out of $\frac{5 (4 + x)}{x - 1}$ .

$f (\frac{5 (4 + x)}{x - 1}) = \frac{5 (\frac{4 + x}{x - 1} + 4)}{5 (\frac{4 + x}{x - 1}) - 5}$

Step 5.3.3.4.2

Factor $5$ out of $- 5$ .

$f (\frac{5 (4 + x)}{x - 1}) = \frac{5 (\frac{4 + x}{x - 1} + 4)}{5 (\frac{4 + x}{x - 1}) + 5 (- 1)}$

Step 5.3.3.4.3

Factor $5$ out of $5 \frac{4 + x}{x - 1} + 5 (- 1)$ .

$f (\frac{5 (4 + x)}{x - 1}) = \frac{5 (\frac{4 + x}{x - 1} + 4)}{5 (\frac{4 + x}{x - 1} - 1)}$

Step 5.3.3.4.4

Cancel the common factor.

$f (\frac{5 (4 + x)}{x - 1}) = \frac{5 (\frac{4 + x}{x - 1} + 4)}{5 (\frac{4 + x}{x - 1} - 1)}$

Step 5.3.3.4.5

Rewrite the expression.

$f (\frac{5 (4 + x)}{x - 1}) = \frac{\frac{4 + x}{x - 1} + 4}{\frac{4 + x}{x - 1} - 1}$

Step 5.3.4

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

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Step 5.3.4.1

Multiply $\frac{\frac{4 + x}{x - 1} + 4}{\frac{4 + x}{x - 1} - 1}$ by $\frac{x - 1}{x - 1}$ .

$f (\frac{5 (4 + x)}{x - 1}) = \frac{x - 1}{x - 1} \cdot \frac{\frac{4 + x}{x - 1} + 4}{\frac{4 + x}{x - 1} - 1}$

Step 5.3.4.2

Combine.

$f (\frac{5 (4 + x)}{x - 1}) = \frac{(x - 1) (\frac{4 + x}{x - 1} + 4)}{(x - 1) (\frac{4 + x}{x - 1} - 1)}$

Step 5.3.5

Apply the distributive property.

$f (\frac{5 (4 + x)}{x - 1}) = \frac{(x - 1) (\frac{4 + x}{x - 1}) + (x - 1) \cdot 4}{(x - 1) (\frac{4 + x}{x - 1}) + (x - 1) \cdot - 1}$

Step 5.3.6

Simplify by cancelling.

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Step 5.3.6.1

Cancel the common factor of $x - 1$ .

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Step 5.3.6.1.1

Cancel the common factor.

$f (\frac{5 (4 + x)}{x - 1}) = \frac{(x - 1) (\frac{4 + x}{x - 1}) + (x - 1) \cdot 4}{(x - 1) (\frac{4 + x}{x - 1}) + (x - 1) \cdot - 1}$

Step 5.3.6.1.2

Rewrite the expression.

$f (\frac{5 (4 + x)}{x - 1}) = \frac{4 + x + (x - 1) \cdot 4}{(x - 1) (\frac{4 + x}{x - 1}) + (x - 1) \cdot - 1}$

Step 5.3.6.2

Cancel the common factor of $x - 1$ .

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Step 5.3.6.2.1

Cancel the common factor.

$f (\frac{5 (4 + x)}{x - 1}) = \frac{4 + x + (x - 1) \cdot 4}{(x - 1) (\frac{4 + x}{x - 1}) + (x - 1) \cdot - 1}$

Step 5.3.6.2.2

Rewrite the expression.

$f (\frac{5 (4 + x)}{x - 1}) = \frac{4 + x + (x - 1) \cdot 4}{4 + x + (x - 1) \cdot - 1}$

Step 5.3.7

Simplify the numerator.

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Step 5.3.7.1

Apply the distributive property.

$f (\frac{5 (4 + x)}{x - 1}) = \frac{4 + x + x \cdot 4 - 1 \cdot 4}{4 + x + (x - 1) \cdot - 1}$

Step 5.3.7.2

Move $4$ to the left of $x$ .

$f (\frac{5 (4 + x)}{x - 1}) = \frac{4 + x + 4 \cdot x - 1 \cdot 4}{4 + x + (x - 1) \cdot - 1}$

Step 5.3.7.3

Multiply $- 1$ by $4$ .

$f (\frac{5 (4 + x)}{x - 1}) = \frac{4 + x + 4 \cdot x - 4}{4 + x + (x - 1) \cdot - 1}$

Step 5.3.7.4

Subtract $4$ from $4$ .

$f (\frac{5 (4 + x)}{x - 1}) = \frac{x + 4 x + 0}{4 + x + (x - 1) \cdot - 1}$

Step 5.3.7.5

Add $x + 4 x$ and $0$ .

$f (\frac{5 (4 + x)}{x - 1}) = \frac{x + 4 x}{4 + x + (x - 1) \cdot - 1}$

Step 5.3.7.6

Add $x$ and $4 x$ .

$f (\frac{5 (4 + x)}{x - 1}) = \frac{5 x}{4 + x + (x - 1) \cdot - 1}$

Step 5.3.8

Simplify the denominator.

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Step 5.3.8.1

Apply the distributive property.

$f (\frac{5 (4 + x)}{x - 1}) = \frac{5 x}{4 + x + x \cdot - 1 - 1 \cdot - 1}$

Step 5.3.8.2

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

$f (\frac{5 (4 + x)}{x - 1}) = \frac{5 x}{4 + x - 1 \cdot x - 1 \cdot - 1}$

Step 5.3.8.3

Multiply $- 1$ by $- 1$ .

$f (\frac{5 (4 + x)}{x - 1}) = \frac{5 x}{4 + x - 1 \cdot x + 1}$

Step 5.3.8.4

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

$f (\frac{5 (4 + x)}{x - 1}) = \frac{5 x}{4 + x - x + 1}$

Step 5.3.8.5

Add $4$ and $1$ .

$f (\frac{5 (4 + x)}{x - 1}) = \frac{5 x}{x - x + 5}$

Step 5.3.8.6

Subtract $x$ from $x$ .

$f (\frac{5 (4 + x)}{x - 1}) = \frac{5 x}{0 + 5}$

Step 5.3.8.7

Add $0$ and $5$ .

$f (\frac{5 (4 + x)}{x - 1}) = \frac{5 x}{5}$

Step 5.3.9

Cancel the common factor of $5$ .

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Step 5.3.9.1

Cancel the common factor.

$f (\frac{5 (4 + x)}{x - 1}) = \frac{5 x}{5}$

Step 5.3.9.2

Divide $x$ by $1$ .

$f (\frac{5 (4 + x)}{x - 1}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{5 (4 + x)}{x - 1}$ is the inverse of $f (x) = \frac{x + 20}{x - 5}$ .

$f^{- 1} (x) = \frac{5 (4 + x)}{x - 1}$