Algebra Examples

$f (x) = \frac{8 + x}{3 x - 2}$

Step 1

Write $f (x) = \frac{8 + x}{3 x - 2}$ as an equation.

$y = \frac{8 + x}{3 x - 2}$

Step 2

Interchange the variables.

$x = \frac{8 + y}{3 y - 2}$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Multiply the equation by $3 y - 2$ .

$(3 y - 2) (x) = (\frac{8 + y}{3 y - 2}) (3 y - 2)$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Apply the distributive property.

$3 y x - 2 x = (\frac{8 + y}{3 y - 2}) (3 y - 2)$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Cancel the common factor of $3 y - 2$ .

Tap for more steps...

Step 3.3.1.1

Cancel the common factor.

$3 y x - 2 x = \frac{8 + y}{3 y - 2} (3 y - 2)$

Step 3.3.1.2

Rewrite the expression.

$3 y x - 2 x = 8 + y$

Step 3.4

Solve for $y$ .

Tap for more steps...

Step 3.4.1

Subtract $y$ from both sides of the equation.

$3 y x - 2 x - y = 8$

Step 3.4.2

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

$3 y x - y = 8 + 2 x$

Step 3.4.3

Factor $y$ out of $3 y x - y$ .

Tap for more steps...

Step 3.4.3.1

Factor $y$ out of $3 y x$ .

$y (3 x) - y = 8 + 2 x$

Step 3.4.3.2

Factor $y$ out of $- y$ .

$y (3 x) + y \cdot - 1 = 8 + 2 x$

Step 3.4.3.3

Factor $y$ out of $y (3 x) + y \cdot - 1$ .

$y (3 x - 1) = 8 + 2 x$

Step 3.4.4

Divide each term in $y (3 x - 1) = 8 + 2 x$ by $3 x - 1$ and simplify.

Tap for more steps...

Step 3.4.4.1

Divide each term in $y (3 x - 1) = 8 + 2 x$ by $3 x - 1$ .

$\frac{y (3 x - 1)}{3 x - 1} = \frac{8}{3 x - 1} + \frac{2 x}{3 x - 1}$

Step 3.4.4.2

Simplify the left side.

Tap for more steps...

Step 3.4.4.2.1

Cancel the common factor of $3 x - 1$ .

Tap for more steps...

Step 3.4.4.2.1.1

Cancel the common factor.

$\frac{y (3 x - 1)}{3 x - 1} = \frac{8}{3 x - 1} + \frac{2 x}{3 x - 1}$

Step 3.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{8}{3 x - 1} + \frac{2 x}{3 x - 1}$

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.

$y = \frac{8 + 2 x}{3 x - 1}$

Step 3.4.4.3.2

Factor $2$ out of $8 + 2 x$ .

Tap for more steps...

Step 3.4.4.3.2.1

Factor $2$ out of $8$ .

$y = \frac{2 \cdot 4 + 2 x}{3 x - 1}$

Step 3.4.4.3.2.2

Factor $2$ out of $2 \cdot 4 + 2 x$ .

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

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = \frac{2 (4 + x)}{3 x - 1}$ is the inverse of $f (x) = \frac{8 + x}{3 x - 2}$ .

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 $f^{- 1} (\frac{8 + x}{3 x - 2})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{2 (4 + \frac{8 + x}{3 x - 2})}{3 (\frac{8 + x}{3 x - 2}) - 1}$

Step 5.2.3

Remove parentheses.

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{2 (4 + \frac{8 + x}{3 x - 2})}{3 (\frac{8 + x}{3 x - 2}) - 1}$

Step 5.2.4

Simplify the numerator.

Tap for more steps...

Step 5.2.4.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{3 x - 2}{3 x - 2}$ .

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{2 (\frac{4 (3 x - 2)}{3 x - 2} + \frac{8 + x}{3 x - 2})}{3 (\frac{8 + x}{3 x - 2}) - 1}$

Step 5.2.4.2

Combine the numerators over the common denominator.

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{2 (\frac{4 (3 x - 2) + 8 + x}{3 x - 2})}{3 (\frac{8 + x}{3 x - 2}) - 1}$

Step 5.2.4.3

Reorder terms.

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{2 (\frac{x + 4 (3 x - 2) + 8}{3 x - 2})}{3 (\frac{8 + x}{3 x - 2}) - 1}$

Step 5.2.4.4

Rewrite $\frac{x + 4 (3 x - 2) + 8}{3 x - 2}$ in a factored form.

Tap for more steps...

Step 5.2.4.4.1

Apply the distributive property.

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{2 (\frac{x + 4 (3 x) + 4 \cdot - 2 + 8}{3 x - 2})}{3 (\frac{8 + x}{3 x - 2}) - 1}$

Step 5.2.4.4.2

Multiply $3$ by $4$ .

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{2 (\frac{x + 12 x + 4 \cdot - 2 + 8}{3 x - 2})}{3 (\frac{8 + x}{3 x - 2}) - 1}$

Step 5.2.4.4.3

Multiply $4$ by $- 2$ .

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{2 (\frac{x + 12 x - 8 + 8}{3 x - 2})}{3 (\frac{8 + x}{3 x - 2}) - 1}$

Step 5.2.4.4.4

Add $x$ and $12 x$ .

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{2 (\frac{13 x - 8 + 8}{3 x - 2})}{3 (\frac{8 + x}{3 x - 2}) - 1}$

Step 5.2.4.4.5

Add $- 8$ and $8$ .

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{2 (\frac{13 x + 0}{3 x - 2})}{3 (\frac{8 + x}{3 x - 2}) - 1}$

Step 5.2.4.4.6

Add $13 x$ and $0$ .

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{2 (\frac{13 x}{3 x - 2})}{3 (\frac{8 + x}{3 x - 2}) - 1}$

Step 5.2.5

Simplify the denominator.

Tap for more steps...

Step 5.2.5.1

Combine $3$ and $\frac{8 + x}{3 x - 2}$ .

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{2 (\frac{13 x}{3 x - 2})}{\frac{3 (8 + x)}{3 x - 2} - 1}$

Step 5.2.5.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3 x - 2}{3 x - 2}$ .

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{2 (\frac{13 x}{3 x - 2})}{\frac{3 (8 + x)}{3 x - 2} - 1 \cdot \frac{3 x - 2}{3 x - 2}}$

Step 5.2.5.3

Combine $- 1$ and $\frac{3 x - 2}{3 x - 2}$ .

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{2 (\frac{13 x}{3 x - 2})}{\frac{3 (8 + x)}{3 x - 2} + \frac{- (3 x - 2)}{3 x - 2}}$

Step 5.2.5.4

Combine the numerators over the common denominator.

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{2 (\frac{13 x}{3 x - 2})}{\frac{3 (8 + x) - (3 x - 2)}{3 x - 2}}$

Step 5.2.5.5

Rewrite $\frac{3 (8 + x) - (3 x - 2)}{3 x - 2}$ in a factored form.

Tap for more steps...

Step 5.2.5.5.1

Apply the distributive property.

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{2 (\frac{13 x}{3 x - 2})}{\frac{3 \cdot 8 + 3 x - (3 x - 2)}{3 x - 2}}$

Step 5.2.5.5.2

Multiply $3$ by $8$ .

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{2 (\frac{13 x}{3 x - 2})}{\frac{24 + 3 x - (3 x - 2)}{3 x - 2}}$

Step 5.2.5.5.3

Apply the distributive property.

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{2 (\frac{13 x}{3 x - 2})}{\frac{24 + 3 x - (3 x) + 2}{3 x - 2}}$

Step 5.2.5.5.4

Multiply $3$ by $- 1$ .

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{2 (\frac{13 x}{3 x - 2})}{\frac{24 + 3 x - 3 x + 2}{3 x - 2}}$

Step 5.2.5.5.5

Multiply $- 1$ by $- 2$ .

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{2 (\frac{13 x}{3 x - 2})}{\frac{24 + 3 x - 3 x + 2}{3 x - 2}}$

Step 5.2.5.5.6

Add $24$ and $2$ .

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{2 (\frac{13 x}{3 x - 2})}{\frac{3 x - 3 x + 26}{3 x - 2}}$

Step 5.2.5.5.7

Subtract $3 x$ from $3 x$ .

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{2 (\frac{13 x}{3 x - 2})}{\frac{0 + 26}{3 x - 2}}$

Step 5.2.5.5.8

Add $0$ and $26$ .

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{2 (\frac{13 x}{3 x - 2})}{\frac{26}{3 x - 2}}$

Step 5.2.6

Combine $2$ and $\frac{13 x}{3 x - 2}$ .

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{\frac{2 (13 x)}{3 x - 2}}{\frac{26}{3 x - 2}}$

Step 5.2.7

Multiply $2$ by $13$ .

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{\frac{26 x}{3 x - 2}}{\frac{26}{3 x - 2}}$

Step 5.2.8

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{26 x}{3 x - 2} \cdot \frac{3 x - 2}{26}$

Step 5.2.9

Cancel the common factor of $26$ .

Tap for more steps...

Step 5.2.9.1

Factor $26$ out of $26 x$ .

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{26 (x)}{3 x - 2} \cdot \frac{3 x - 2}{26}$

Step 5.2.9.2

Cancel the common factor.

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{26 x}{3 x - 2} \cdot \frac{3 x - 2}{26}$

Step 5.2.9.3

Rewrite the expression.

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{x}{3 x - 2} \cdot (3 x - 2)$

Step 5.2.10

Cancel the common factor of $3 x - 2$ .

Tap for more steps...

Step 5.2.10.1

Cancel the common factor.

$f^{- 1} (\frac{8 + x}{3 x - 2}) = \frac{x}{3 x - 2} \cdot (3 x - 2)$

Step 5.2.10.2

Rewrite the expression.

$f^{- 1} (\frac{8 + x}{3 x - 2}) = x$

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 $f (\frac{2 (4 + x)}{3 x - 1})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{8 + \frac{2 (4 + x)}{3 x - 1}}{3 (\frac{2 (4 + x)}{3 x - 1}) - 2}$

Step 5.3.3

Remove parentheses.

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{8 + \frac{2 (4 + x)}{3 x - 1}}{3 (\frac{2 (4 + x)}{3 x - 1}) - 2}$

Step 5.3.4

Multiply the numerator and denominator of the fraction by $3 x - 1$ .

Tap for more steps...

Step 5.3.4.1

Multiply $\frac{8 + \frac{2 (4 + x)}{3 x - 1}}{3 \frac{2 (4 + x)}{3 x - 1} - 2}$ by $\frac{3 x - 1}{3 x - 1}$ .

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{3 x - 1}{3 x - 1} \cdot \frac{8 + \frac{2 (4 + x)}{3 x - 1}}{3 (\frac{2 (4 + x)}{3 x - 1}) - 2}$

Step 5.3.4.2

Combine.

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{(3 x - 1) (8 + \frac{2 (4 + x)}{3 x - 1})}{(3 x - 1) (3 (\frac{2 (4 + x)}{3 x - 1}) - 2)}$

Step 5.3.5

Apply the distributive property.

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{(3 x - 1) \cdot 8 + (3 x - 1) (\frac{2 (4 + x)}{3 x - 1})}{(3 x - 1) (3 (\frac{2 (4 + x)}{3 x - 1})) + (3 x - 1) \cdot - 2}$

Step 5.3.6

Cancel the common factor of $3 x - 1$ .

Tap for more steps...

Step 5.3.6.1

Cancel the common factor.

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{(3 x - 1) \cdot 8 + (3 x - 1) (\frac{2 (4 + x)}{3 x - 1})}{(3 x - 1) (3 (\frac{2 (4 + x)}{3 x - 1})) + (3 x - 1) \cdot - 2}$

Step 5.3.6.2

Rewrite the expression.

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{(3 x - 1) \cdot 8 + 2 (4 + x)}{(3 x - 1) (3 (\frac{2 (4 + x)}{3 x - 1})) + (3 x - 1) \cdot - 2}$

Step 5.3.7

Simplify the numerator.

Tap for more steps...

Step 5.3.7.1

Factor $2$ out of $(3 x - 1) \cdot 8 + 2 (4 + x)$ .

Tap for more steps...

Step 5.3.7.1.1

Factor $2$ out of $(3 x - 1) \cdot 8$ .

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

Step 5.3.7.1.2

Factor $2$ out of $2 ((3 x - 1) \cdot 4) + 2 (4 + x)$ .

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

Step 5.3.7.2

Apply the distributive property.

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

Step 5.3.7.3

Multiply $4$ by $3$ .

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{2 (12 x - 1 \cdot 4 + 4 + x)}{(3 x - 1) \cdot (3 (\frac{2 (4 + x)}{3 x - 1})) + (3 x - 1) \cdot - 2}$

Step 5.3.7.4

Multiply $- 1$ by $4$ .

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{2 (12 x - 4 + 4 + x)}{(3 x - 1) \cdot (3 (\frac{2 (4 + x)}{3 x - 1})) + (3 x - 1) \cdot - 2}$

Step 5.3.7.5

Add $12 x$ and $x$ .

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{2 (13 x - 4 + 4)}{(3 x - 1) \cdot (3 (\frac{2 (4 + x)}{3 x - 1})) + (3 x - 1) \cdot - 2}$

Step 5.3.7.6

Add $- 4$ and $4$ .

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{2 (13 x + 0)}{(3 x - 1) \cdot (3 (\frac{2 (4 + x)}{3 x - 1})) + (3 x - 1) \cdot - 2}$

Step 5.3.7.7

Add $13 x$ and $0$ .

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{2 \cdot (13 x)}{(3 x - 1) \cdot (3 (\frac{2 (4 + x)}{3 x - 1})) + (3 x - 1) \cdot - 2}$

Step 5.3.7.8

Multiply $2$ by $13$ .

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{26 x}{(3 x - 1) \cdot (3 (\frac{2 (4 + x)}{3 x - 1})) + (3 x - 1) \cdot - 2}$

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{26 x}{(3 x - 1) (3 (\frac{2 (4 + x)}{3 x - 1})) + (3 x - 1) \cdot - 2}$

Step 5.3.8

Simplify the denominator.

Tap for more steps...

Step 5.3.8.1

Factor $3 x - 1$ out of $(3 x - 1) \cdot 3 \frac{2 (4 + x)}{3 x - 1} + (3 x - 1) \cdot - 2$ .

Tap for more steps...

Step 5.3.8.1.1

Factor $3 x - 1$ out of $(3 x - 1) \cdot - 2$ .

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{26 x}{(3 x - 1) (3 (\frac{2 (4 + x)}{3 x - 1})) + (3 x - 1) (- 2)}$

Step 5.3.8.1.2

Factor $3 x - 1$ out of $(3 x - 1) (3 \frac{2 (4 + x)}{3 x - 1}) + (3 x - 1) (- 2)$ .

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{26 x}{(3 x - 1) (3 (\frac{2 (4 + x)}{3 x - 1}) - 2)}$

Step 5.3.8.2

Multiply $3 \frac{2 (4 + x)}{3 x - 1}$ .

Tap for more steps...

Step 5.3.8.2.1

Combine $3$ and $\frac{2 (4 + x)}{3 x - 1}$ .

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{26 x}{(3 x - 1) (\frac{3 (2 (4 + x))}{3 x - 1} - 2)}$

Step 5.3.8.2.2

Multiply $2$ by $3$ .

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{26 x}{(3 x - 1) (\frac{6 (4 + x)}{3 x - 1} - 2)}$

Step 5.3.8.3

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{3 x - 1}{3 x - 1}$ .

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{26 x}{(3 x - 1) (\frac{6 (4 + x)}{3 x - 1} - 2 \cdot \frac{3 x - 1}{3 x - 1})}$

Step 5.3.8.4

Combine $- 2$ and $\frac{3 x - 1}{3 x - 1}$ .

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{26 x}{(3 x - 1) (\frac{6 (4 + x)}{3 x - 1} + \frac{- 2 (3 x - 1)}{3 x - 1})}$

Step 5.3.8.5

Combine the numerators over the common denominator.

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{26 x}{(3 x - 1) (\frac{6 (4 + x) - 2 (3 x - 1)}{3 x - 1})}$

Step 5.3.8.6

Rewrite $\frac{6 (4 + x) - 2 (3 x - 1)}{3 x - 1}$ in a factored form.

Tap for more steps...

Step 5.3.8.6.1

Factor $2$ out of $6 (4 + x) - 2 (3 x - 1)$ .

Tap for more steps...

Step 5.3.8.6.1.1

Factor $2$ out of $6 (4 + x)$ .

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{26 x}{(3 x - 1) (\frac{2 (3 (4 + x)) - 2 (3 x - 1)}{3 x - 1})}$

Step 5.3.8.6.1.2

Factor $2$ out of $- 2 (3 x - 1)$ .

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{26 x}{(3 x - 1) (\frac{2 (3 (4 + x)) + 2 (- (3 x - 1))}{3 x - 1})}$

Step 5.3.8.6.1.3

Factor $2$ out of $2 (3 (4 + x)) + 2 (- (3 x - 1))$ .

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{26 x}{(3 x - 1) (\frac{2 (3 (4 + x) - (3 x - 1))}{3 x - 1})}$

Step 5.3.8.6.2

Apply the distributive property.

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{26 x}{(3 x - 1) (\frac{2 (3 \cdot 4 + 3 x - (3 x - 1))}{3 x - 1})}$

Step 5.3.8.6.3

Multiply $3$ by $4$ .

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{26 x}{(3 x - 1) (\frac{2 (12 + 3 x - (3 x - 1))}{3 x - 1})}$

Step 5.3.8.6.4

Apply the distributive property.

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{26 x}{(3 x - 1) (\frac{2 (12 + 3 x - (3 x) + 1)}{3 x - 1})}$

Step 5.3.8.6.5

Multiply $3$ by $- 1$ .

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{26 x}{(3 x - 1) (\frac{2 (12 + 3 x - 3 x + 1)}{3 x - 1})}$

Step 5.3.8.6.6

Multiply $- 1$ by $- 1$ .

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{26 x}{(3 x - 1) (\frac{2 (12 + 3 x - 3 x + 1)}{3 x - 1})}$

Step 5.3.8.6.7

Add $12$ and $1$ .

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{26 x}{(3 x - 1) (\frac{2 (3 x - 3 x + 13)}{3 x - 1})}$

Step 5.3.8.6.8

Subtract $3 x$ from $3 x$ .

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{26 x}{(3 x - 1) (\frac{2 (0 + 13)}{3 x - 1})}$

Step 5.3.8.6.9

Add $0$ and $13$ .

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{26 x}{(3 x - 1) (\frac{2 \cdot 13}{3 x - 1})}$

Step 5.3.8.7

Multiply $2$ by $13$ .

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{26 x}{(3 x - 1) (\frac{26}{3 x - 1})}$

Step 5.3.9

Factor $\frac{26}{3 x - 1}$ out of $\frac{26 x}{(3 x - 1) \frac{26}{3 x - 1}}$ .

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{3 x - 1}{26} \cdot \frac{26 x}{3 x - 1}$

Step 5.3.10

Cancel the common factor of $3 x - 1$ .

Tap for more steps...

Step 5.3.10.1

Cancel the common factor.

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{3 x - 1}{26} \cdot \frac{26 x}{3 x - 1}$

Step 5.3.10.2

Rewrite the expression.

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{1}{26} \cdot (26 x)$

Step 5.3.11

Cancel the common factor of $26$ .

Tap for more steps...

Step 5.3.11.1

Factor $26$ out of $26 x$ .

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{1}{26} \cdot (26 (x))$

Step 5.3.11.2

Cancel the common factor.

$f (\frac{2 (4 + x)}{3 x - 1}) = \frac{1}{26} \cdot (26 x)$

Step 5.3.11.3

Rewrite the expression.

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

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{2 (4 + x)}{3 x - 1}$ is the inverse of $f (x) = \frac{8 + x}{3 x - 2}$ .

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