Precalculus Examples

$f (x) = \frac{11 x - 13}{- 2 x + 5}$

Step 1

Write $f (x) = \frac{11 x - 13}{- 2 x + 5}$ as an equation.

$y = \frac{11 x - 13}{- 2 x + 5}$

Step 2

Interchange the variables.

$x = \frac{11 y - 13}{- 2 y + 5}$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $\frac{11 y - 13}{- 2 y + 5} = x$ .

$\frac{11 y - 13}{- 2 y + 5} = x$

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.

$- 2 y + 5, 1$

Step 3.2.2

Remove parentheses.

$- 2 y + 5, 1$

Step 3.2.3

The LCM of one and any expression is the expression.

$- 2 y + 5$

Step 3.3

Multiply each term in $\frac{11 y - 13}{- 2 y + 5} = x$ by $- 2 y + 5$ to eliminate the fractions.

Tap for more steps...

Step 3.3.1

Multiply each term in $\frac{11 y - 13}{- 2 y + 5} = x$ by $- 2 y + 5$ .

$\frac{11 y - 13}{- 2 y + 5} (- 2 y + 5) = x (- 2 y + 5)$

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Multiply $\frac{11 y - 13}{- 2 y + 5}$ by $- 2 y + 5$ .

$\frac{(11 y - 13) (- 2 y + 5)}{- 2 y + 5} = x (- 2 y + 5)$

Step 3.3.2.2

Cancel the common factor of $- 2 y + 5$ .

Tap for more steps...

Step 3.3.2.2.1

Cancel the common factor.

$\frac{(11 y - 13) (- 2 y + 5)}{- 2 y + 5} = x (- 2 y + 5)$

Step 3.3.2.2.2

Divide $11 y - 13$ by $1$ .

$11 y - 13 = x (- 2 y + 5)$

Step 3.3.3

Simplify the right side.

Tap for more steps...

Step 3.3.3.1

Apply the distributive property.

$11 y - 13 = x (- 2 y) + x \cdot 5$

Step 3.3.3.2

Reorder.

Tap for more steps...

Step 3.3.3.2.1

Rewrite using the commutative property of multiplication.

$11 y - 13 = - 2 x y + x \cdot 5$

Step 3.3.3.2.2

Move $5$ to the left of $x$ .

$11 y - 13 = - 2 x y + 5 x$

Step 3.4

Solve the equation.

Tap for more steps...

Step 3.4.1

Add $2 x y$ to both sides of the equation.

$11 y - 13 + 2 x y = 5 x$

Step 3.4.2

Add $13$ to both sides of the equation.

$11 y + 2 x y = 5 x + 13$

Step 3.4.3

Factor $y$ out of $11 y + 2 x y$ .

Tap for more steps...

Step 3.4.3.1

Factor $y$ out of $11 y$ .

$y \cdot 11 + 2 x y = 5 x + 13$

Step 3.4.3.2

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

$y \cdot 11 + y (2 x) = 5 x + 13$

Step 3.4.3.3

Factor $y$ out of $y \cdot 11 + y (2 x)$ .

$y (11 + 2 x) = 5 x + 13$

Step 3.4.4

Divide each term in $y (11 + 2 x) = 5 x + 13$ by $11 + 2 x$ and simplify.

Tap for more steps...

Step 3.4.4.1

Divide each term in $y (11 + 2 x) = 5 x + 13$ by $11 + 2 x$ .

$\frac{y (11 + 2 x)}{11 + 2 x} = \frac{5 x}{11 + 2 x} + \frac{13}{11 + 2 x}$

Step 3.4.4.2

Simplify the left side.

Tap for more steps...

Step 3.4.4.2.1

Cancel the common factor of $11 + 2 x$ .

Tap for more steps...

Step 3.4.4.2.1.1

Cancel the common factor.

$\frac{y (11 + 2 x)}{11 + 2 x} = \frac{5 x}{11 + 2 x} + \frac{13}{11 + 2 x}$

Step 3.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{5 x}{11 + 2 x} + \frac{13}{11 + 2 x}$

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.

$y = \frac{5 x + 13}{11 + 2 x}$

Step 4

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

$f^{- 1} (x) = \frac{5 x + 13}{11 + 2 x}$

Step 5

Verify if $f^{- 1} (x) = \frac{5 x + 13}{11 + 2 x}$ is the inverse of $f (x) = \frac{11 x - 13}{- 2 x + 5}$ .

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 $f^{- 1} (\frac{11 x - 13}{- 2 x + 5})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{5 (\frac{11 x - 13}{- 2 x + 5}) + 13}{11 + 2 (\frac{11 x - 13}{- 2 x + 5})}$

Step 5.2.3

Simplify the numerator.

Tap for more steps...

Step 5.2.3.1

Combine $5$ and $\frac{11 x - 13}{- 2 x + 5}$ .

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{\frac{5 (11 x - 13)}{- 2 x + 5} + 13}{11 + 2 (\frac{11 x - 13}{- 2 x + 5})}$

Step 5.2.3.2

To write $13$ as a fraction with a common denominator, multiply by $\frac{- 2 x + 5}{- 2 x + 5}$ .

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{\frac{5 (11 x - 13)}{- 2 x + 5} + \frac{13 (- 2 x + 5)}{- 2 x + 5}}{11 + 2 (\frac{11 x - 13}{- 2 x + 5})}$

Step 5.2.3.3

Combine the numerators over the common denominator.

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{\frac{5 (11 x - 13) + 13 (- 2 x + 5)}{- 2 x + 5}}{11 + 2 (\frac{11 x - 13}{- 2 x + 5})}$

Step 5.2.3.4

Reorder terms.

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{\frac{13 (- 2 x + 5) + 5 (11 x - 13)}{- 2 x + 5}}{11 + 2 (\frac{11 x - 13}{- 2 x + 5})}$

Step 5.2.3.5

Rewrite $\frac{13 (- 2 x + 5) + 5 (11 x - 13)}{- 2 x + 5}$ in a factored form.

Tap for more steps...

Step 5.2.3.5.1

Apply the distributive property.

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{\frac{13 (- 2 x) + 13 \cdot 5 + 5 (11 x - 13)}{- 2 x + 5}}{11 + 2 (\frac{11 x - 13}{- 2 x + 5})}$

Step 5.2.3.5.2

Multiply $- 2$ by $13$ .

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{\frac{- 26 x + 13 \cdot 5 + 5 (11 x - 13)}{- 2 x + 5}}{11 + 2 (\frac{11 x - 13}{- 2 x + 5})}$

Step 5.2.3.5.3

Multiply $13$ by $5$ .

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{\frac{- 26 x + 65 + 5 (11 x - 13)}{- 2 x + 5}}{11 + 2 (\frac{11 x - 13}{- 2 x + 5})}$

Step 5.2.3.5.4

Apply the distributive property.

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{\frac{- 26 x + 65 + 5 (11 x) + 5 \cdot - 13}{- 2 x + 5}}{11 + 2 (\frac{11 x - 13}{- 2 x + 5})}$

Step 5.2.3.5.5

Multiply $11$ by $5$ .

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{\frac{- 26 x + 65 + 55 x + 5 \cdot - 13}{- 2 x + 5}}{11 + 2 (\frac{11 x - 13}{- 2 x + 5})}$

Step 5.2.3.5.6

Multiply $5$ by $- 13$ .

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{\frac{- 26 x + 65 + 55 x - 65}{- 2 x + 5}}{11 + 2 (\frac{11 x - 13}{- 2 x + 5})}$

Step 5.2.3.5.7

Add $- 26 x$ and $55 x$ .

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{\frac{29 x + 65 - 65}{- 2 x + 5}}{11 + 2 (\frac{11 x - 13}{- 2 x + 5})}$

Step 5.2.3.5.8

Subtract $65$ from $65$ .

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{\frac{29 x + 0}{- 2 x + 5}}{11 + 2 (\frac{11 x - 13}{- 2 x + 5})}$

Step 5.2.3.5.9

Add $29 x$ and $0$ .

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{\frac{29 x}{- 2 x + 5}}{11 + 2 (\frac{11 x - 13}{- 2 x + 5})}$

Step 5.2.4

Simplify the denominator.

Tap for more steps...

Step 5.2.4.1

Combine $2$ and $\frac{11 x - 13}{- 2 x + 5}$ .

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{\frac{29 x}{- 2 x + 5}}{11 + \frac{2 (11 x - 13)}{- 2 x + 5}}$

Step 5.2.4.2

To write $11$ as a fraction with a common denominator, multiply by $\frac{- 2 x + 5}{- 2 x + 5}$ .

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{\frac{29 x}{- 2 x + 5}}{\frac{11 (- 2 x + 5)}{- 2 x + 5} + \frac{2 (11 x - 13)}{- 2 x + 5}}$

Step 5.2.4.3

Combine the numerators over the common denominator.

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{\frac{29 x}{- 2 x + 5}}{\frac{11 (- 2 x + 5) + 2 (11 x - 13)}{- 2 x + 5}}$

Step 5.2.4.4

Rewrite $\frac{11 (- 2 x + 5) + 2 (11 x - 13)}{- 2 x + 5}$ in a factored form.

Tap for more steps...

Step 5.2.4.4.1

Apply the distributive property.

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{\frac{29 x}{- 2 x + 5}}{\frac{11 (- 2 x) + 11 \cdot 5 + 2 (11 x - 13)}{- 2 x + 5}}$

Step 5.2.4.4.2

Multiply $- 2$ by $11$ .

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{\frac{29 x}{- 2 x + 5}}{\frac{- 22 x + 11 \cdot 5 + 2 (11 x - 13)}{- 2 x + 5}}$

Step 5.2.4.4.3

Multiply $11$ by $5$ .

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{\frac{29 x}{- 2 x + 5}}{\frac{- 22 x + 55 + 2 (11 x - 13)}{- 2 x + 5}}$

Step 5.2.4.4.4

Apply the distributive property.

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{\frac{29 x}{- 2 x + 5}}{\frac{- 22 x + 55 + 2 (11 x) + 2 \cdot - 13}{- 2 x + 5}}$

Step 5.2.4.4.5

Multiply $11$ by $2$ .

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{\frac{29 x}{- 2 x + 5}}{\frac{- 22 x + 55 + 22 x + 2 \cdot - 13}{- 2 x + 5}}$

Step 5.2.4.4.6

Multiply $2$ by $- 13$ .

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{\frac{29 x}{- 2 x + 5}}{\frac{- 22 x + 55 + 22 x - 26}{- 2 x + 5}}$

Step 5.2.4.4.7

Add $- 22 x$ and $22 x$ .

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{\frac{29 x}{- 2 x + 5}}{\frac{0 + 55 - 26}{- 2 x + 5}}$

Step 5.2.4.4.8

Add $0$ and $55$ .

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{\frac{29 x}{- 2 x + 5}}{\frac{55 - 26}{- 2 x + 5}}$

Step 5.2.4.4.9

Subtract $26$ from $55$ .

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{\frac{29 x}{- 2 x + 5}}{\frac{29}{- 2 x + 5}}$

Step 5.2.5

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{29 x}{- 2 x + 5} \cdot \frac{- 2 x + 5}{29}$

Step 5.2.6

Cancel the common factor of $29$ .

Tap for more steps...

Step 5.2.6.1

Factor $29$ out of $29 x$ .

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{29 (x)}{- 2 x + 5} \cdot \frac{- 2 x + 5}{29}$

Step 5.2.6.2

Cancel the common factor.

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{29 x}{- 2 x + 5} \cdot \frac{- 2 x + 5}{29}$

Step 5.2.6.3

Rewrite the expression.

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{x}{- 2 x + 5} \cdot (- 2 x + 5)$

Step 5.2.7

Multiply $\frac{x}{- 2 x + 5}$ by $- 2 x + 5$ .

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{x (- 2 x + 5)}{- 2 x + 5}$

Step 5.2.8

Cancel the common factor of $- 2 x + 5$ .

Tap for more steps...

Step 5.2.8.1

Cancel the common factor.

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = \frac{x (- 2 x + 5)}{- 2 x + 5}$

Step 5.2.8.2

Divide $x$ by $1$ .

$f^{- 1} (\frac{11 x - 13}{- 2 x + 5}) = x$

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 $f (\frac{5 x + 13}{11 + 2 x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{11 (\frac{5 x + 13}{11 + 2 x}) - 13}{- 2 \frac{5 x + 13}{11 + 2 x} + 5}$

Step 5.3.3

Simplify the numerator.

Tap for more steps...

Step 5.3.3.1

Combine $11$ and $\frac{5 x + 13}{11 + 2 x}$ .

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{\frac{11 (5 x + 13)}{11 + 2 x} - 13}{- 2 \frac{5 x + 13}{11 + 2 x} + 5}$

Step 5.3.3.2

To write $- 13$ as a fraction with a common denominator, multiply by $\frac{11 + 2 x}{11 + 2 x}$ .

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{\frac{11 (5 x + 13)}{11 + 2 x} - 13 \cdot \frac{11 + 2 x}{11 + 2 x}}{- 2 \frac{5 x + 13}{11 + 2 x} + 5}$

Step 5.3.3.3

Combine $- 13$ and $\frac{11 + 2 x}{11 + 2 x}$ .

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{\frac{11 (5 x + 13)}{11 + 2 x} + \frac{- 13 (11 + 2 x)}{11 + 2 x}}{- 2 \frac{5 x + 13}{11 + 2 x} + 5}$

Step 5.3.3.4

Combine the numerators over the common denominator.

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{\frac{11 (5 x + 13) - 13 (11 + 2 x)}{11 + 2 x}}{- 2 \frac{5 x + 13}{11 + 2 x} + 5}$

Step 5.3.3.5

Reorder terms.

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{\frac{11 (5 x + 13) - 13 (11 + 2 x)}{2 x + 11}}{- 2 \frac{5 x + 13}{11 + 2 x} + 5}$

Step 5.3.3.6

Rewrite $\frac{11 (5 x + 13) - 13 (11 + 2 x)}{2 x + 11}$ in a factored form.

Tap for more steps...

Step 5.3.3.6.1

Apply the distributive property.

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{\frac{11 (5 x) + 11 \cdot 13 - 13 (11 + 2 x)}{2 x + 11}}{- 2 \frac{5 x + 13}{11 + 2 x} + 5}$

Step 5.3.3.6.2

Multiply $5$ by $11$ .

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{\frac{55 x + 11 \cdot 13 - 13 (11 + 2 x)}{2 x + 11}}{- 2 \frac{5 x + 13}{11 + 2 x} + 5}$

Step 5.3.3.6.3

Multiply $11$ by $13$ .

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{\frac{55 x + 143 - 13 (11 + 2 x)}{2 x + 11}}{- 2 \frac{5 x + 13}{11 + 2 x} + 5}$

Step 5.3.3.6.4

Apply the distributive property.

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{\frac{55 x + 143 - 13 \cdot 11 - 13 (2 x)}{2 x + 11}}{- 2 \frac{5 x + 13}{11 + 2 x} + 5}$

Step 5.3.3.6.5

Multiply $- 13$ by $11$ .

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{\frac{55 x + 143 - 143 - 13 (2 x)}{2 x + 11}}{- 2 \frac{5 x + 13}{11 + 2 x} + 5}$

Step 5.3.3.6.6

Multiply $2$ by $- 13$ .

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{\frac{55 x + 143 - 143 - 26 x}{2 x + 11}}{- 2 \frac{5 x + 13}{11 + 2 x} + 5}$

Step 5.3.3.6.7

Subtract $26 x$ from $55 x$ .

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{\frac{29 x + 143 - 143}{2 x + 11}}{- 2 \frac{5 x + 13}{11 + 2 x} + 5}$

Step 5.3.3.6.8

Subtract $143$ from $143$ .

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{\frac{29 x + 0}{2 x + 11}}{- 2 \frac{5 x + 13}{11 + 2 x} + 5}$

Step 5.3.3.6.9

Add $29 x$ and $0$ .

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{\frac{29 x}{2 x + 11}}{- 2 \frac{5 x + 13}{11 + 2 x} + 5}$

Step 5.3.4

Simplify the denominator.

Tap for more steps...

Step 5.3.4.1

Combine $- 2$ and $\frac{5 x + 13}{11 + 2 x}$ .

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{\frac{29 x}{2 x + 11}}{\frac{- 2 (5 x + 13)}{11 + 2 x} + 5}$

Step 5.3.4.2

Move the negative in front of the fraction.

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{\frac{29 x}{2 x + 11}}{- \frac{2 (5 x + 13)}{11 + 2 x} + 5}$

Step 5.3.4.3

To write $5$ as a fraction with a common denominator, multiply by $\frac{11 + 2 x}{11 + 2 x}$ .

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{\frac{29 x}{2 x + 11}}{- \frac{2 (5 x + 13)}{11 + 2 x} + \frac{5 (11 + 2 x)}{11 + 2 x}}$

Step 5.3.4.4

Combine the numerators over the common denominator.

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{\frac{29 x}{2 x + 11}}{\frac{- 2 (5 x + 13) + 5 (11 + 2 x)}{11 + 2 x}}$

Step 5.3.4.5

Rewrite $\frac{- 2 (5 x + 13) + 5 (11 + 2 x)}{11 + 2 x}$ in a factored form.

Tap for more steps...

Step 5.3.4.5.1

Apply the distributive property.

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{\frac{29 x}{2 x + 11}}{\frac{- 2 (5 x) - 2 \cdot 13 + 5 (11 + 2 x)}{11 + 2 x}}$

Step 5.3.4.5.2

Multiply $5$ by $- 2$ .

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{\frac{29 x}{2 x + 11}}{\frac{- 10 x - 2 \cdot 13 + 5 (11 + 2 x)}{11 + 2 x}}$

Step 5.3.4.5.3

Multiply $- 2$ by $13$ .

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{\frac{29 x}{2 x + 11}}{\frac{- 10 x - 26 + 5 (11 + 2 x)}{11 + 2 x}}$

Step 5.3.4.5.4

Apply the distributive property.

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{\frac{29 x}{2 x + 11}}{\frac{- 10 x - 26 + 5 \cdot 11 + 5 (2 x)}{11 + 2 x}}$

Step 5.3.4.5.5

Multiply $5$ by $11$ .

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{\frac{29 x}{2 x + 11}}{\frac{- 10 x - 26 + 55 + 5 (2 x)}{11 + 2 x}}$

Step 5.3.4.5.6

Multiply $2$ by $5$ .

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{\frac{29 x}{2 x + 11}}{\frac{- 10 x - 26 + 55 + 10 x}{11 + 2 x}}$

Step 5.3.4.5.7

Add $- 10 x$ and $10 x$ .

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{\frac{29 x}{2 x + 11}}{\frac{0 - 26 + 55}{11 + 2 x}}$

Step 5.3.4.5.8

Subtract $26$ from $0$ .

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{\frac{29 x}{2 x + 11}}{\frac{- 26 + 55}{11 + 2 x}}$

Step 5.3.4.5.9

Add $- 26$ and $55$ .

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{\frac{29 x}{2 x + 11}}{\frac{29}{11 + 2 x}}$

Step 5.3.5

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{29 x}{2 x + 11} \cdot \frac{11 + 2 x}{29}$

Step 5.3.6

Cancel the common factor of $29$ .

Tap for more steps...

Step 5.3.6.1

Factor $29$ out of $29 x$ .

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{29 (x)}{2 x + 11} \cdot \frac{11 + 2 x}{29}$

Step 5.3.6.2

Cancel the common factor.

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{29 x}{2 x + 11} \cdot \frac{11 + 2 x}{29}$

Step 5.3.6.3

Rewrite the expression.

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{x}{2 x + 11} \cdot (11 + 2 x)$

Step 5.3.7

Multiply $\frac{x}{2 x + 11}$ by $11 + 2 x$ .

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{x (11 + 2 x)}{2 x + 11}$

Step 5.3.8

Cancel the common factor of $11 + 2 x$ and $2 x + 11$ .

Tap for more steps...

Step 5.3.8.1

Reorder terms.

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{x (2 x + 11)}{2 x + 11}$

Step 5.3.8.2

Cancel the common factor.

$f (\frac{5 x + 13}{11 + 2 x}) = \frac{x (2 x + 11)}{2 x + 11}$

Step 5.3.8.3

Divide $x$ by $1$ .

$f (\frac{5 x + 13}{11 + 2 x}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{5 x + 13}{11 + 2 x}$ is the inverse of $f (x) = \frac{11 x - 13}{- 2 x + 5}$ .

$f^{- 1} (x) = \frac{5 x + 13}{11 + 2 x}$