Precalculus Examples

$f (x) = \frac{2 x}{x - 6}$

Step 1

Write $f (x) = \frac{2 x}{x - 6}$ as an equation.

$y = \frac{2 x}{x - 6}$

Step 2

Interchange the variables.

$x = \frac{2 y}{y - 6}$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Multiply the equation by $y - 6$ .

$(y - 6) (x) = (\frac{2 y}{y - 6}) (y - 6)$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Apply the distributive property.

$y x - 6 x = (\frac{2 y}{y - 6}) (y - 6)$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Cancel the common factor of $y - 6$ .

Tap for more steps...

Step 3.3.1.1

Cancel the common factor.

$y x - 6 x = \frac{2 y}{y - 6} (y - 6)$

Step 3.3.1.2

Rewrite the expression.

$y x - 6 x = 2 y$

Step 3.4

Solve for $y$ .

Tap for more steps...

Step 3.4.1

Subtract $2 y$ from both sides of the equation.

$y x - 6 x - 2 y = 0$

Step 3.4.2

Add $6 x$ to both sides of the equation.

$y x - 2 y = 6 x$

Step 3.4.3

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

Tap for more steps...

Step 3.4.3.1

Factor $y$ out of $y x$ .

$y (x) - 2 y = 6 x$

Step 3.4.3.2

Factor $y$ out of $- 2 y$ .

$y (x) + y \cdot - 2 = 6 x$

Step 3.4.3.3

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

$y (x - 2) = 6 x$

Step 3.4.4

Divide each term in $y (x - 2) = 6 x$ by $x - 2$ and simplify.

Tap for more steps...

Step 3.4.4.1

Divide each term in $y (x - 2) = 6 x$ by $x - 2$ .

$\frac{y (x - 2)}{x - 2} = \frac{6 x}{x - 2}$

Step 3.4.4.2

Simplify the left side.

Tap for more steps...

Step 3.4.4.2.1

Cancel the common factor of $x - 2$ .

Tap for more steps...

Step 3.4.4.2.1.1

Cancel the common factor.

$\frac{y (x - 2)}{x - 2} = \frac{6 x}{x - 2}$

Step 3.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{6 x}{x - 2}$

Step 4

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

$f^{- 1} (x) = \frac{6 x}{x - 2}$

Step 5

Verify if $f^{- 1} (x) = \frac{6 x}{x - 2}$ is the inverse of $f (x) = \frac{2 x}{x - 6}$ .

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 $f^{- 1} (\frac{2 x}{x - 6})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{2 x}{x - 6}) = \frac{6 (\frac{2 x}{x - 6})}{(\frac{2 x}{x - 6}) - 2}$

Step 5.2.3

Simplify terms.

Tap for more steps...

Step 5.2.3.1

Cancel the common factor of $6$ and $(\frac{2 x}{x - 6}) - 2$ .

Tap for more steps...

Step 5.2.3.1.1

Factor $2$ out of $6 (\frac{2 x}{x - 6})$ .

$f^{- 1} (\frac{2 x}{x - 6}) = \frac{2 (3 (\frac{2 x}{x - 6}))}{(\frac{2 x}{x - 6}) - 2}$

Step 5.2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 5.2.3.1.2.1

Factor $2$ out of $\frac{2 x}{x - 6}$ .

$f^{- 1} (\frac{2 x}{x - 6}) = \frac{2 (3 (\frac{2 x}{x - 6}))}{2 (\frac{x}{x - 6}) - 2}$

Step 5.2.3.1.2.2

Factor $2$ out of $- 2$ .

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

Step 5.2.3.1.2.3

Factor $2$ out of $2 \frac{x}{x - 6} + 2 (- 1)$ .

$f^{- 1} (\frac{2 x}{x - 6}) = \frac{2 (3 (\frac{2 x}{x - 6}))}{2 (\frac{x}{x - 6} - 1)}$

Step 5.2.3.1.2.4

Cancel the common factor.

$f^{- 1} (\frac{2 x}{x - 6}) = \frac{2 (3 (\frac{2 x}{x - 6}))}{2 (\frac{x}{x - 6} - 1)}$

Step 5.2.3.1.2.5

Rewrite the expression.

$f^{- 1} (\frac{2 x}{x - 6}) = \frac{3 (\frac{2 x}{x - 6})}{\frac{x}{x - 6} - 1}$

Step 5.2.3.2

Combine $3$ and $\frac{2 x}{x - 6}$ .

$f^{- 1} (\frac{2 x}{x - 6}) = \frac{\frac{3 (2 x)}{x - 6}}{\frac{x}{x - 6} - 1}$

Step 5.2.4

Simplify the denominator.

Tap for more steps...

Step 5.2.4.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x - 6}{x - 6}$ .

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

Step 5.2.4.2

Combine $- 1$ and $\frac{x - 6}{x - 6}$ .

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

Step 5.2.4.3

Combine the numerators over the common denominator.

$f^{- 1} (\frac{2 x}{x - 6}) = \frac{\frac{3 (2 x)}{x - 6}}{\frac{x - (x - 6)}{x - 6}}$

Step 5.2.4.4

Rewrite $\frac{x - (x - 6)}{x - 6}$ in a factored form.

Tap for more steps...

Step 5.2.4.4.1

Apply the distributive property.

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

Step 5.2.4.4.2

Multiply $- 1$ by $- 6$ .

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

Step 5.2.4.4.3

Subtract $x$ from $x$ .

$f^{- 1} (\frac{2 x}{x - 6}) = \frac{\frac{3 (2 x)}{x - 6}}{\frac{0 + 6}{x - 6}}$

Step 5.2.4.4.4

Add $0$ and $6$ .

$f^{- 1} (\frac{2 x}{x - 6}) = \frac{\frac{3 (2 x)}{x - 6}}{\frac{6}{x - 6}}$

Step 5.2.5

Multiply $3$ by $2$ .

$f^{- 1} (\frac{2 x}{x - 6}) = \frac{\frac{6 x}{x - 6}}{\frac{6}{x - 6}}$

Step 5.2.6

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{2 x}{x - 6}) = \frac{6 x}{x - 6} \cdot \frac{x - 6}{6}$

Step 5.2.7

Cancel the common factor of $6$ .

Tap for more steps...

Step 5.2.7.1

Factor $6$ out of $6 x$ .

$f^{- 1} (\frac{2 x}{x - 6}) = \frac{6 (x)}{x - 6} \cdot \frac{x - 6}{6}$

Step 5.2.7.2

Cancel the common factor.

$f^{- 1} (\frac{2 x}{x - 6}) = \frac{6 x}{x - 6} \cdot \frac{x - 6}{6}$

Step 5.2.7.3

Rewrite the expression.

$f^{- 1} (\frac{2 x}{x - 6}) = \frac{x}{x - 6} \cdot (x - 6)$

Step 5.2.8

Cancel the common factor of $x - 6$ .

Tap for more steps...

Step 5.2.8.1

Cancel the common factor.

$f^{- 1} (\frac{2 x}{x - 6}) = \frac{x}{x - 6} \cdot (x - 6)$

Step 5.2.8.2

Rewrite the expression.

$f^{- 1} (\frac{2 x}{x - 6}) = x$

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 $f (\frac{6 x}{x - 2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{6 x}{x - 2}) = \frac{2 (\frac{6 x}{x - 2})}{(\frac{6 x}{x - 2}) - 6}$

Step 5.3.3

Cancel the common factors.

Tap for more steps...

Step 5.3.3.1

Factor $2$ out of $\frac{6 x}{x - 2}$ .

$f (\frac{6 x}{x - 2}) = \frac{2 (\frac{6 x}{x - 2})}{2 (\frac{3 x}{x - 2}) - 6}$

Step 5.3.3.2

Factor $2$ out of $- 6$ .

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

Step 5.3.3.3

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

$f (\frac{6 x}{x - 2}) = \frac{2 (\frac{6 x}{x - 2})}{2 (\frac{3 x}{x - 2} - 3)}$

Step 5.3.3.4

Cancel the common factor.

$f (\frac{6 x}{x - 2}) = \frac{2 (\frac{6 x}{x - 2})}{2 (\frac{3 x}{x - 2} - 3)}$

Step 5.3.3.5

Rewrite the expression.

$f (\frac{6 x}{x - 2}) = \frac{\frac{6 x}{x - 2}}{\frac{3 x}{x - 2} - 3}$

Step 5.3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.5

Simplify the denominator.

Tap for more steps...

Step 5.3.5.1

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

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

Step 5.3.5.2

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

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

Step 5.3.5.3

Combine the numerators over the common denominator.

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

Step 5.3.5.4

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

Tap for more steps...

Step 5.3.5.4.1

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

Tap for more steps...

Step 5.3.5.4.1.1

Factor $3$ out of $3 x$ .

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

Step 5.3.5.4.1.2

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

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

Step 5.3.5.4.1.3

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

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

Step 5.3.5.4.2

Apply the distributive property.

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

Step 5.3.5.4.3

Multiply $- 1$ by $- 2$ .

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

Step 5.3.5.4.4

Subtract $x$ from $x$ .

$f (\frac{6 x}{x - 2}) = \frac{6 x}{x - 2} \cdot \frac{1}{\frac{3 (0 + 2)}{x - 2}}$

Step 5.3.5.4.5

Add $0$ and $2$ .

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

Step 5.3.5.5

Multiply $3$ by $2$ .

$f (\frac{6 x}{x - 2}) = \frac{6 x}{x - 2} \cdot \frac{1}{\frac{6}{x - 2}}$

Step 5.3.6

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{6 x}{x - 2}) = \frac{6 x}{x - 2} \cdot (1 (\frac{x - 2}{6}))$

Step 5.3.7

Multiply $\frac{x - 2}{6}$ by $1$ .

$f (\frac{6 x}{x - 2}) = \frac{6 x}{x - 2} \cdot \frac{x - 2}{6}$

Step 5.3.8

Cancel the common factor of $6$ .

Tap for more steps...

Step 5.3.8.1

Factor $6$ out of $6 x$ .

$f (\frac{6 x}{x - 2}) = \frac{6 (x)}{x - 2} \cdot \frac{x - 2}{6}$

Step 5.3.8.2

Cancel the common factor.

$f (\frac{6 x}{x - 2}) = \frac{6 x}{x - 2} \cdot \frac{x - 2}{6}$

Step 5.3.8.3

Rewrite the expression.

$f (\frac{6 x}{x - 2}) = \frac{x}{x - 2} \cdot (x - 2)$

Step 5.3.9

Cancel the common factor of $x - 2$ .

Tap for more steps...

Step 5.3.9.1

Cancel the common factor.

$f (\frac{6 x}{x - 2}) = \frac{x}{x - 2} \cdot (x - 2)$

Step 5.3.9.2

Rewrite the expression.

$f (\frac{6 x}{x - 2}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{6 x}{x - 2}$ is the inverse of $f (x) = \frac{2 x}{x - 6}$ .

$f^{- 1} (x) = \frac{6 x}{x - 2}$