Precalculus Examples

$f (x) = \frac{x + 5}{x + 10}$

Step 1

Write $f (x) = \frac{x + 5}{x + 10}$ as an equation.

$y = \frac{x + 5}{x + 10}$

Step 2

Interchange the variables.

$x = \frac{y + 5}{y + 10}$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $\frac{y + 5}{y + 10} = x$ .

$\frac{y + 5}{y + 10} = x$

Step 3.2

Find the LCD of the terms in the equation.

Tap for more steps...

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.

$y + 10, 1$

Step 3.2.2

Remove parentheses.

$y + 10, 1$

Step 3.2.3

The LCM of one and any expression is the expression.

$y + 10$

Step 3.3

Multiply each term in $\frac{y + 5}{y + 10} = x$ by $y + 10$ to eliminate the fractions.

Tap for more steps...

Step 3.3.1

Multiply each term in $\frac{y + 5}{y + 10} = x$ by $y + 10$ .

$\frac{y + 5}{y + 10} (y + 10) = x (y + 10)$

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Cancel the common factor of $y + 10$ .

Tap for more steps...

Step 3.3.2.1.1

Cancel the common factor.

$\frac{y + 5}{y + 10} (y + 10) = x (y + 10)$

Step 3.3.2.1.2

Rewrite the expression.

$y + 5 = x (y + 10)$

Step 3.3.3

Simplify the right side.

Tap for more steps...

Step 3.3.3.1

Apply the distributive property.

$y + 5 = x y + x \cdot 10$

Step 3.3.3.2

Move $10$ to the left of $x$ .

$y + 5 = x y + 10 x$

Step 3.4

Solve the equation.

Tap for more steps...

Step 3.4.1

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

$y + 5 - x y = 10 x$

Step 3.4.2

Subtract $5$ from both sides of the equation.

$y - x y = 10 x - 5$

Step 3.4.3

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

Tap for more steps...

Step 3.4.3.1

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

$y \cdot 1 - x y = 10 x - 5$

Step 3.4.3.2

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

$y \cdot 1 + y (- x) = 10 x - 5$

Step 3.4.3.3

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

$y (1 - x) = 10 x - 5$

Step 3.4.4

Divide each term in $y (1 - x) = 10 x - 5$ by $1 - x$ and simplify.

Tap for more steps...

Step 3.4.4.1

Divide each term in $y (1 - x) = 10 x - 5$ by $1 - x$ .

$\frac{y (1 - x)}{1 - x} = \frac{10 x}{1 - x} + \frac{- 5}{1 - x}$

Step 3.4.4.2

Simplify the left side.

Tap for more steps...

Step 3.4.4.2.1

Cancel the common factor of $1 - x$ .

Tap for more steps...

Step 3.4.4.2.1.1

Cancel the common factor.

$\frac{y (1 - x)}{1 - x} = \frac{10 x}{1 - x} + \frac{- 5}{1 - x}$

Step 3.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{10 x}{1 - x} + \frac{- 5}{1 - x}$

Step 3.4.4.3

Simplify the right side.

Tap for more steps...

Step 3.4.4.3.1

Combine the numerators over the common denominator.

$y = \frac{10 x - 5}{1 - x}$

Step 3.4.4.3.2

Factor $5$ out of $10 x - 5$ .

Tap for more steps...

Step 3.4.4.3.2.1

Factor $5$ out of $10 x$ .

$y = \frac{5 (2 x) - 5}{1 - x}$

Step 3.4.4.3.2.2

Factor $5$ out of $- 5$ .

$y = \frac{5 (2 x) + 5 (- 1)}{1 - x}$

Step 3.4.4.3.2.3

Factor $5$ out of $5 (2 x) + 5 (- 1)$ .

$y = \frac{5 (2 x - 1)}{1 - x}$

Step 4

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

$f^{- 1} (x) = \frac{5 (2 x - 1)}{1 - x}$

Step 5

Verify if $f^{- 1} (x) = \frac{5 (2 x - 1)}{1 - x}$ is the inverse of $f (x) = \frac{x + 5}{x + 10}$ .

Tap for more steps...

Step 5.1

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (\frac{x + 5}{x + 10})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x + 5}{x + 10}) = \frac{5 (2 (\frac{x + 5}{x + 10}) - 1)}{1 - (\frac{x + 5}{x + 10})}$

Step 5.2.3

Simplify the numerator.

Tap for more steps...

Step 5.2.3.1

Combine $2$ and $\frac{x + 5}{x + 10}$ .

$f^{- 1} (\frac{x + 5}{x + 10}) = \frac{5 (\frac{2 (x + 5)}{x + 10} - 1)}{1 - \frac{x + 5}{x + 10}}$

Step 5.2.3.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x + 10}{x + 10}$ .

$f^{- 1} (\frac{x + 5}{x + 10}) = \frac{5 (\frac{2 (x + 5)}{x + 10} - 1 \cdot \frac{x + 10}{x + 10})}{1 - \frac{x + 5}{x + 10}}$

Step 5.2.3.3

Combine $- 1$ and $\frac{x + 10}{x + 10}$ .

$f^{- 1} (\frac{x + 5}{x + 10}) = \frac{5 (\frac{2 (x + 5)}{x + 10} + \frac{- (x + 10)}{x + 10})}{1 - \frac{x + 5}{x + 10}}$

Step 5.2.3.4

Combine the numerators over the common denominator.

$f^{- 1} (\frac{x + 5}{x + 10}) = \frac{5 (\frac{2 (x + 5) - (x + 10)}{x + 10})}{1 - \frac{x + 5}{x + 10}}$

Step 5.2.3.5

Rewrite $\frac{2 (x + 5) - (x + 10)}{x + 10}$ in a factored form.

Tap for more steps...

Step 5.2.3.5.1

Apply the distributive property.

$f^{- 1} (\frac{x + 5}{x + 10}) = \frac{5 (\frac{2 x + 2 \cdot 5 - (x + 10)}{x + 10})}{1 - \frac{x + 5}{x + 10}}$

Step 5.2.3.5.2

Multiply $2$ by $5$ .

$f^{- 1} (\frac{x + 5}{x + 10}) = \frac{5 (\frac{2 x + 10 - (x + 10)}{x + 10})}{1 - \frac{x + 5}{x + 10}}$

Step 5.2.3.5.3

Apply the distributive property.

$f^{- 1} (\frac{x + 5}{x + 10}) = \frac{5 (\frac{2 x + 10 - x - 1 \cdot 10}{x + 10})}{1 - \frac{x + 5}{x + 10}}$

Step 5.2.3.5.4

Multiply $- 1$ by $10$ .

$f^{- 1} (\frac{x + 5}{x + 10}) = \frac{5 (\frac{2 x + 10 - x - 10}{x + 10})}{1 - \frac{x + 5}{x + 10}}$

Step 5.2.3.5.5

Subtract $x$ from $2 x$ .

$f^{- 1} (\frac{x + 5}{x + 10}) = \frac{5 (\frac{x + 10 - 10}{x + 10})}{1 - \frac{x + 5}{x + 10}}$

Step 5.2.3.5.6

Subtract $10$ from $10$ .

$f^{- 1} (\frac{x + 5}{x + 10}) = \frac{5 (\frac{x + 0}{x + 10})}{1 - \frac{x + 5}{x + 10}}$

Step 5.2.3.5.7

Add $x$ and $0$ .

$f^{- 1} (\frac{x + 5}{x + 10}) = \frac{5 (\frac{x}{x + 10})}{1 - \frac{x + 5}{x + 10}}$

Step 5.2.4

Simplify the denominator.

Tap for more steps...

Step 5.2.4.1

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

$f^{- 1} (\frac{x + 5}{x + 10}) = \frac{5 (\frac{x}{x + 10})}{\frac{x + 10}{x + 10} - \frac{x + 5}{x + 10}}$

Step 5.2.4.2

Combine the numerators over the common denominator.

$f^{- 1} (\frac{x + 5}{x + 10}) = \frac{5 (\frac{x}{x + 10})}{\frac{x + 10 - (x + 5)}{x + 10}}$

Step 5.2.4.3

Rewrite $\frac{x + 10 - (x + 5)}{x + 10}$ in a factored form.

Tap for more steps...

Step 5.2.4.3.1

Apply the distributive property.

$f^{- 1} (\frac{x + 5}{x + 10}) = \frac{5 (\frac{x}{x + 10})}{\frac{x + 10 - x - 1 \cdot 5}{x + 10}}$

Step 5.2.4.3.2

Multiply $- 1$ by $5$ .

$f^{- 1} (\frac{x + 5}{x + 10}) = \frac{5 (\frac{x}{x + 10})}{\frac{x + 10 - x - 5}{x + 10}}$

Step 5.2.4.3.3

Subtract $x$ from $x$ .

$f^{- 1} (\frac{x + 5}{x + 10}) = \frac{5 (\frac{x}{x + 10})}{\frac{0 + 10 - 5}{x + 10}}$

Step 5.2.4.3.4

Add $0$ and $10$ .

$f^{- 1} (\frac{x + 5}{x + 10}) = \frac{5 (\frac{x}{x + 10})}{\frac{10 - 5}{x + 10}}$

Step 5.2.4.3.5

Subtract $5$ from $10$ .

$f^{- 1} (\frac{x + 5}{x + 10}) = \frac{5 (\frac{x}{x + 10})}{\frac{5}{x + 10}}$

Step 5.2.5

Combine $5$ and $\frac{x}{x + 10}$ .

$f^{- 1} (\frac{x + 5}{x + 10}) = \frac{\frac{5 x}{x + 10}}{\frac{5}{x + 10}}$

Step 5.2.6

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{x + 5}{x + 10}) = \frac{5 x}{x + 10} \cdot \frac{x + 10}{5}$

Step 5.2.7

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.2.7.1

Factor $5$ out of $5 x$ .

$f^{- 1} (\frac{x + 5}{x + 10}) = \frac{5 (x)}{x + 10} \cdot \frac{x + 10}{5}$

Step 5.2.7.2

Cancel the common factor.

$f^{- 1} (\frac{x + 5}{x + 10}) = \frac{5 x}{x + 10} \cdot \frac{x + 10}{5}$

Step 5.2.7.3

Rewrite the expression.

$f^{- 1} (\frac{x + 5}{x + 10}) = \frac{x}{x + 10} \cdot (x + 10)$

Step 5.2.8

Cancel the common factor of $x + 10$ .

Tap for more steps...

Step 5.2.8.1

Cancel the common factor.

$f^{- 1} (\frac{x + 5}{x + 10}) = \frac{x}{x + 10} \cdot (x + 10)$

Step 5.2.8.2

Rewrite the expression.

$f^{- 1} (\frac{x + 5}{x + 10}) = x$

Step 5.3

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

Tap for more steps...

Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (\frac{5 (2 x - 1)}{1 - x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{5 (2 x - 1)}{1 - x}) = \frac{(\frac{5 (2 x - 1)}{1 - x}) + 5}{(\frac{5 (2 x - 1)}{1 - x}) + 10}$

Step 5.3.3

Cancel the common factor of $(\frac{5 (2 x - 1)}{1 - x}) + 5$ and $(\frac{5 (2 x - 1)}{1 - x}) + 10$ .

Tap for more steps...

Step 5.3.3.1

Factor $5$ out of $\frac{5 (2 x - 1)}{1 - x}$ .

$f (\frac{5 (2 x - 1)}{1 - x}) = \frac{5 (\frac{2 x - 1}{1 - x}) + 5}{(\frac{5 (2 x - 1)}{1 - x}) + 10}$

Step 5.3.3.2

Factor $5$ out of $5$ .

$f (\frac{5 (2 x - 1)}{1 - x}) = \frac{5 (\frac{2 x - 1}{1 - x}) + 5 \cdot 1}{\frac{5 (2 x - 1)}{1 - x} + 10}$

Step 5.3.3.3

Factor $5$ out of $5 \frac{2 x - 1}{1 - x} + 5 (1)$ .

$f (\frac{5 (2 x - 1)}{1 - x}) = \frac{5 (\frac{2 x - 1}{1 - x} + 1)}{(\frac{5 (2 x - 1)}{1 - x}) + 10}$

Step 5.3.3.4

Cancel the common factors.

Tap for more steps...

Step 5.3.3.4.1

Factor $5$ out of $\frac{5 (2 x - 1)}{1 - x}$ .

$f (\frac{5 (2 x - 1)}{1 - x}) = \frac{5 (\frac{2 x - 1}{1 - x} + 1)}{5 (\frac{2 x - 1}{1 - x}) + 10}$

Step 5.3.3.4.2

Factor $5$ out of $10$ .

$f (\frac{5 (2 x - 1)}{1 - x}) = \frac{5 (\frac{2 x - 1}{1 - x} + 1)}{5 (\frac{2 x - 1}{1 - x}) + 5 \cdot 2}$

Step 5.3.3.4.3

Factor $5$ out of $5 \frac{2 x - 1}{1 - x} + 5 \cdot 2$ .

$f (\frac{5 (2 x - 1)}{1 - x}) = \frac{5 (\frac{2 x - 1}{1 - x} + 1)}{5 (\frac{2 x - 1}{1 - x} + 2)}$

Step 5.3.3.4.4

Cancel the common factor.

$f (\frac{5 (2 x - 1)}{1 - x}) = \frac{5 (\frac{2 x - 1}{1 - x} + 1)}{5 (\frac{2 x - 1}{1 - x} + 2)}$

Step 5.3.3.4.5

Rewrite the expression.

$f (\frac{5 (2 x - 1)}{1 - x}) = \frac{\frac{2 x - 1}{1 - x} + 1}{\frac{2 x - 1}{1 - x} + 2}$

Step 5.3.4

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

Tap for more steps...

Step 5.3.4.1

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

$f (\frac{5 (2 x - 1)}{1 - x}) = \frac{1 - x}{1 - x} \cdot \frac{\frac{2 x - 1}{1 - x} + 1}{\frac{2 x - 1}{1 - x} + 2}$

Step 5.3.4.2

Combine.

$f (\frac{5 (2 x - 1)}{1 - x}) = \frac{(1 - x) (\frac{2 x - 1}{1 - x} + 1)}{(1 - x) (\frac{2 x - 1}{1 - x} + 2)}$

Step 5.3.5

Apply the distributive property.

$f (\frac{5 (2 x - 1)}{1 - x}) = \frac{(1 - x) (\frac{2 x - 1}{1 - x}) + (1 - x) \cdot 1}{(1 - x) (\frac{2 x - 1}{1 - x}) + (1 - x) \cdot 2}$

Step 5.3.6

Simplify by cancelling.

Tap for more steps...

Step 5.3.6.1

Cancel the common factor of $1 - x$ .

Tap for more steps...

Step 5.3.6.1.1

Cancel the common factor.

$f (\frac{5 (2 x - 1)}{1 - x}) = \frac{(1 - x) (\frac{2 x - 1}{1 - x}) + (1 - x) \cdot 1}{(1 - x) (\frac{2 x - 1}{1 - x}) + (1 - x) \cdot 2}$

Step 5.3.6.1.2

Rewrite the expression.

$f (\frac{5 (2 x - 1)}{1 - x}) = \frac{2 x - 1 + (1 - x) \cdot 1}{(1 - x) (\frac{2 x - 1}{1 - x}) + (1 - x) \cdot 2}$

Step 5.3.6.2

Cancel the common factor of $1 - x$ .

Tap for more steps...

Step 5.3.6.2.1

Cancel the common factor.

$f (\frac{5 (2 x - 1)}{1 - x}) = \frac{2 x - 1 + (1 - x) \cdot 1}{(1 - x) (\frac{2 x - 1}{1 - x}) + (1 - x) \cdot 2}$

Step 5.3.6.2.2

Rewrite the expression.

$f (\frac{5 (2 x - 1)}{1 - x}) = \frac{2 x - 1 + (1 - x) \cdot 1}{2 x - 1 + (1 - x) \cdot 2}$

Step 5.3.7

Simplify the numerator.

Tap for more steps...

Step 5.3.7.1

Multiply $1 - x$ by $1$ .

$f (\frac{5 (2 x - 1)}{1 - x}) = \frac{2 x - 1 + 1 - x}{2 x - 1 + (1 - x) \cdot 2}$

Step 5.3.7.2

Subtract $x$ from $2 x$ .

$f (\frac{5 (2 x - 1)}{1 - x}) = \frac{x - 1 + 1}{2 x - 1 + (1 - x) \cdot 2}$

Step 5.3.7.3

Add $- 1$ and $1$ .

$f (\frac{5 (2 x - 1)}{1 - x}) = \frac{x + 0}{2 x - 1 + (1 - x) \cdot 2}$

Step 5.3.7.4

Add $x$ and $0$ .

$f (\frac{5 (2 x - 1)}{1 - x}) = \frac{x}{2 x - 1 + (1 - x) \cdot 2}$

Step 5.3.8

Simplify the denominator.

Tap for more steps...

Step 5.3.8.1

Apply the distributive property.

$f (\frac{5 (2 x - 1)}{1 - x}) = \frac{x}{2 x - 1 + 1 \cdot 2 - x \cdot 2}$

Step 5.3.8.2

Multiply $2$ by $1$ .

$f (\frac{5 (2 x - 1)}{1 - x}) = \frac{x}{2 x - 1 + 2 - x \cdot 2}$

Step 5.3.8.3

Multiply $2$ by $- 1$ .

$f (\frac{5 (2 x - 1)}{1 - x}) = \frac{x}{2 x - 1 + 2 - 2 x}$

Step 5.3.8.4

Subtract $2 x$ from $2 x$ .

$f (\frac{5 (2 x - 1)}{1 - x}) = \frac{x}{0 - 1 + 2}$

Step 5.3.8.5

Subtract $1$ from $0$ .

$f (\frac{5 (2 x - 1)}{1 - x}) = \frac{x}{- 1 + 2}$

Step 5.3.8.6

Add $- 1$ and $2$ .

$f (\frac{5 (2 x - 1)}{1 - x}) = \frac{x}{1}$

Step 5.3.9

Divide $x$ by $1$ .

$f (\frac{5 (2 x - 1)}{1 - x}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{5 (2 x - 1)}{1 - x}$ is the inverse of $f (x) = \frac{x + 5}{x + 10}$ .

$f^{- 1} (x) = \frac{5 (2 x - 1)}{1 - x}$