Precalculus Examples

$f (x) = \frac{6 x}{3 x - 5}$

Step 1

Write $f (x) = \frac{6 x}{3 x - 5}$ as an equation.

$y = \frac{6 x}{3 x - 5}$

Step 2

Interchange the variables.

$x = \frac{6 y}{3 y - 5}$

Step 3

Solve for $y$ .

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Step 3.1

Multiply the equation by $3 y - 5$ .

$(3 y - 5) (x) = (\frac{6 y}{3 y - 5}) (3 y - 5)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Apply the distributive property.

$3 y x - 5 x = (\frac{6 y}{3 y - 5}) (3 y - 5)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $3 y - 5$ .

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Step 3.3.1.1

Cancel the common factor.

$3 y x - 5 x = \frac{6 y}{3 y - 5} (3 y - 5)$

Step 3.3.1.2

Rewrite the expression.

$3 y x - 5 x = 6 y$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $6 y$ from both sides of the equation.

$3 y x - 5 x - 6 y = 0$

Step 3.4.2

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

$3 y x - 6 y = 5 x$

Step 3.4.3

Factor $3 y$ out of $3 y x - 6 y$ .

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Step 3.4.3.1

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

$3 y (x) - 6 y = 5 x$

Step 3.4.3.2

Factor $3 y$ out of $- 6 y$ .

$3 y (x) + 3 y (- 2) = 5 x$

Step 3.4.3.3

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

$3 y (x - 2) = 5 x$

Step 3.4.4

Divide each term in $3 y (x - 2) = 5 x$ by $3 (x - 2)$ and simplify.

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Step 3.4.4.1

Divide each term in $3 y (x - 2) = 5 x$ by $3 (x - 2)$ .

$\frac{3 y (x - 2)}{3 (x - 2)} = \frac{5 x}{3 (x - 2)}$

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

Cancel the common factor of $3$ .

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Step 3.4.4.2.1.1

Cancel the common factor.

$\frac{3 y (x - 2)}{3 (x - 2)} = \frac{5 x}{3 (x - 2)}$

Step 3.4.4.2.1.2

Rewrite the expression.

$\frac{y (x - 2)}{x - 2} = \frac{5 x}{3 (x - 2)}$

Step 3.4.4.2.2

Cancel the common factor of $x - 2$ .

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Step 3.4.4.2.2.1

Cancel the common factor.

$\frac{y (x - 2)}{x - 2} = \frac{5 x}{3 (x - 2)}$

Step 3.4.4.2.2.2

Divide $y$ by $1$ .

$y = \frac{5 x}{3 (x - 2)}$

Step 4

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

$f^{- 1} (x) = \frac{5 x}{3 (x - 2)}$

Step 5

Verify if $f^{- 1} (x) = \frac{5 x}{3 (x - 2)}$ is the inverse of $f (x) = \frac{6 x}{3 x - 5}$ .

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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 $f^{- 1} (\frac{6 x}{3 x - 5})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{6 x}{3 x - 5}) = \frac{5 (\frac{6 x}{3 x - 5})}{3 ((\frac{6 x}{3 x - 5}) - 2)}$

Step 5.2.3

Combine $5$ and $\frac{6 x}{3 x - 5}$ .

$f^{- 1} (\frac{6 x}{3 x - 5}) = \frac{\frac{5 (6 x)}{3 x - 5}}{3 (\frac{6 x}{3 x - 5} - 2)}$

Step 5.2.4

Simplify the denominator.

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Step 5.2.4.1

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{3 x - 5}{3 x - 5}$ .

$f^{- 1} (\frac{6 x}{3 x - 5}) = \frac{\frac{5 (6 x)}{3 x - 5}}{3 (\frac{6 x}{3 x - 5} - 2 \cdot \frac{3 x - 5}{3 x - 5})}$

Step 5.2.4.2

Combine $- 2$ and $\frac{3 x - 5}{3 x - 5}$ .

$f^{- 1} (\frac{6 x}{3 x - 5}) = \frac{\frac{5 (6 x)}{3 x - 5}}{3 (\frac{6 x}{3 x - 5} + \frac{- 2 (3 x - 5)}{3 x - 5})}$

Step 5.2.4.3

Combine the numerators over the common denominator.

$f^{- 1} (\frac{6 x}{3 x - 5}) = \frac{\frac{5 (6 x)}{3 x - 5}}{3 (\frac{6 x - 2 (3 x - 5)}{3 x - 5})}$

Step 5.2.4.4

Rewrite $\frac{6 x - 2 (3 x - 5)}{3 x - 5}$ in a factored form.

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Step 5.2.4.4.1

Factor $2$ out of $6 x - 2 (3 x - 5)$ .

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Step 5.2.4.4.1.1

Factor $2$ out of $6 x$ .

$f^{- 1} (\frac{6 x}{3 x - 5}) = \frac{\frac{5 (6 x)}{3 x - 5}}{3 (\frac{2 (3 x) - 2 (3 x - 5)}{3 x - 5})}$

Step 5.2.4.4.1.2

Factor $2$ out of $- 2 (3 x - 5)$ .

$f^{- 1} (\frac{6 x}{3 x - 5}) = \frac{\frac{5 (6 x)}{3 x - 5}}{3 (\frac{2 (3 x) + 2 (- (3 x - 5))}{3 x - 5})}$

Step 5.2.4.4.1.3

Factor $2$ out of $2 (3 x) + 2 (- (3 x - 5))$ .

$f^{- 1} (\frac{6 x}{3 x - 5}) = \frac{\frac{5 (6 x)}{3 x - 5}}{3 (\frac{2 (3 x - (3 x - 5))}{3 x - 5})}$

Step 5.2.4.4.2

Apply the distributive property.

$f^{- 1} (\frac{6 x}{3 x - 5}) = \frac{\frac{5 (6 x)}{3 x - 5}}{3 (\frac{2 (3 x - (3 x) + 5)}{3 x - 5})}$

Step 5.2.4.4.3

Multiply $3$ by $- 1$ .

$f^{- 1} (\frac{6 x}{3 x - 5}) = \frac{\frac{5 (6 x)}{3 x - 5}}{3 (\frac{2 (3 x - 3 x + 5)}{3 x - 5})}$

Step 5.2.4.4.4

Multiply $- 1$ by $- 5$ .

$f^{- 1} (\frac{6 x}{3 x - 5}) = \frac{\frac{5 (6 x)}{3 x - 5}}{3 (\frac{2 (3 x - 3 x + 5)}{3 x - 5})}$

Step 5.2.4.4.5

Subtract $3 x$ from $3 x$ .

$f^{- 1} (\frac{6 x}{3 x - 5}) = \frac{\frac{5 (6 x)}{3 x - 5}}{3 (\frac{2 (0 + 5)}{3 x - 5})}$

Step 5.2.4.4.6

Add $0$ and $5$ .

$f^{- 1} (\frac{6 x}{3 x - 5}) = \frac{\frac{5 (6 x)}{3 x - 5}}{3 (\frac{2 \cdot 5}{3 x - 5})}$

Step 5.2.4.5

Multiply $2$ by $5$ .

$f^{- 1} (\frac{6 x}{3 x - 5}) = \frac{\frac{5 (6 x)}{3 x - 5}}{3 (\frac{10}{3 x - 5})}$

Step 5.2.5

Multiply $5$ by $6$ .

$f^{- 1} (\frac{6 x}{3 x - 5}) = \frac{\frac{30 x}{3 x - 5}}{3 (\frac{10}{3 x - 5})}$

Step 5.2.6

Combine fractions.

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Step 5.2.6.1

Combine $3$ and $\frac{10}{3 x - 5}$ .

$f^{- 1} (\frac{6 x}{3 x - 5}) = \frac{\frac{30 x}{3 x - 5}}{\frac{3 \cdot 10}{3 x - 5}}$

Step 5.2.6.2

Multiply $3$ by $10$ .

$f^{- 1} (\frac{6 x}{3 x - 5}) = \frac{\frac{30 x}{3 x - 5}}{\frac{30}{3 x - 5}}$

Step 5.2.7

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{6 x}{3 x - 5}) = \frac{30 x}{3 x - 5} \cdot \frac{3 x - 5}{30}$

Step 5.2.8

Cancel the common factor of $30$ .

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Step 5.2.8.1

Factor $30$ out of $30 x$ .

$f^{- 1} (\frac{6 x}{3 x - 5}) = \frac{30 (x)}{3 x - 5} \cdot \frac{3 x - 5}{30}$

Step 5.2.8.2

Cancel the common factor.

$f^{- 1} (\frac{6 x}{3 x - 5}) = \frac{30 x}{3 x - 5} \cdot \frac{3 x - 5}{30}$

Step 5.2.8.3

Rewrite the expression.

$f^{- 1} (\frac{6 x}{3 x - 5}) = \frac{x}{3 x - 5} \cdot (3 x - 5)$

Step 5.2.9

Cancel the common factor of $3 x - 5$ .

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Step 5.2.9.1

Cancel the common factor.

$f^{- 1} (\frac{6 x}{3 x - 5}) = \frac{x}{3 x - 5} \cdot (3 x - 5)$

Step 5.2.9.2

Rewrite the expression.

$f^{- 1} (\frac{6 x}{3 x - 5}) = x$

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 $f (\frac{5 x}{3 (x - 2)})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{5 x}{3 (x - 2)}) = \frac{6 (\frac{5 x}{3 (x - 2)})}{3 (\frac{5 x}{3 (x - 2)}) - 5}$

Step 5.3.3

Combine $6$ and $\frac{5 x}{3 (x - 2)}$ .

$f (\frac{5 x}{3 (x - 2)}) = \frac{\frac{6 (5 x)}{3 (x - 2)}}{3 (\frac{5 x}{3 (x - 2)}) - 5}$

Step 5.3.4

Simplify the denominator.

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Step 5.3.4.1

Cancel the common factor of $3$ .

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Step 5.3.4.1.1

Cancel the common factor.

$f (\frac{5 x}{3 (x - 2)}) = \frac{\frac{6 (5 x)}{3 (x - 2)}}{3 (\frac{5 x}{3 (x - 2)}) - 5}$

Step 5.3.4.1.2

Rewrite the expression.

$f (\frac{5 x}{3 (x - 2)}) = \frac{\frac{6 (5 x)}{3 (x - 2)}}{\frac{5 x}{x - 2} - 5}$

Step 5.3.4.2

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

$f (\frac{5 x}{3 (x - 2)}) = \frac{\frac{6 (5 x)}{3 (x - 2)}}{\frac{5 x}{x - 2} - 5 \cdot \frac{x - 2}{x - 2}}$

Step 5.3.4.3

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

$f (\frac{5 x}{3 (x - 2)}) = \frac{\frac{6 (5 x)}{3 (x - 2)}}{\frac{5 x}{x - 2} + \frac{- 5 (x - 2)}{x - 2}}$

Step 5.3.4.4

Combine the numerators over the common denominator.

$f (\frac{5 x}{3 (x - 2)}) = \frac{\frac{6 (5 x)}{3 (x - 2)}}{\frac{5 x - 5 (x - 2)}{x - 2}}$

Step 5.3.4.5

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

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Step 5.3.4.5.1

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

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Step 5.3.4.5.1.1

Factor $5$ out of $5 x$ .

$f (\frac{5 x}{3 (x - 2)}) = \frac{\frac{6 (5 x)}{3 (x - 2)}}{\frac{5 (x) - 5 (x - 2)}{x - 2}}$

Step 5.3.4.5.1.2

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

$f (\frac{5 x}{3 (x - 2)}) = \frac{\frac{6 (5 x)}{3 (x - 2)}}{\frac{5 (x) + 5 (- (x - 2))}{x - 2}}$

Step 5.3.4.5.1.3

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

$f (\frac{5 x}{3 (x - 2)}) = \frac{\frac{6 (5 x)}{3 (x - 2)}}{\frac{5 (x - (x - 2))}{x - 2}}$

Step 5.3.4.5.2

Apply the distributive property.

$f (\frac{5 x}{3 (x - 2)}) = \frac{\frac{6 (5 x)}{3 (x - 2)}}{\frac{5 (x - x + 2)}{x - 2}}$

Step 5.3.4.5.3

Multiply $- 1$ by $- 2$ .

$f (\frac{5 x}{3 (x - 2)}) = \frac{\frac{6 (5 x)}{3 (x - 2)}}{\frac{5 (x - x + 2)}{x - 2}}$

Step 5.3.4.5.4

Subtract $x$ from $x$ .

$f (\frac{5 x}{3 (x - 2)}) = \frac{\frac{6 (5 x)}{3 (x - 2)}}{\frac{5 (0 + 2)}{x - 2}}$

Step 5.3.4.5.5

Add $0$ and $2$ .

$f (\frac{5 x}{3 (x - 2)}) = \frac{\frac{6 (5 x)}{3 (x - 2)}}{\frac{5 \cdot 2}{x - 2}}$

Step 5.3.4.6

Multiply $5$ by $2$ .

$f (\frac{5 x}{3 (x - 2)}) = \frac{\frac{6 (5 x)}{3 (x - 2)}}{\frac{10}{x - 2}}$

Step 5.3.5

Multiply $6$ by $5$ .

$f (\frac{5 x}{3 (x - 2)}) = \frac{\frac{30 x}{3 (x - 2)}}{\frac{10}{x - 2}}$

Step 5.3.6

Reduce the expression $\frac{30 x}{3 (x - 2)}$ by cancelling the common factors.

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Step 5.3.6.1

Factor $3$ out of $30 x$ .

$f (\frac{5 x}{3 (x - 2)}) = \frac{\frac{3 (10 x)}{3 (x - 2)}}{\frac{10}{x - 2}}$

Step 5.3.6.2

Cancel the common factor.

$f (\frac{5 x}{3 (x - 2)}) = \frac{\frac{3 (10 x)}{3 (x - 2)}}{\frac{10}{x - 2}}$

Step 5.3.6.3

Rewrite the expression.

$f (\frac{5 x}{3 (x - 2)}) = \frac{\frac{10 x}{x - 2}}{\frac{10}{x - 2}}$

Step 5.3.7

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{5 x}{3 (x - 2)}) = \frac{10 x}{x - 2} \cdot \frac{x - 2}{10}$

Step 5.3.8

Cancel the common factor of $10$ .

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Step 5.3.8.1

Factor $10$ out of $10 x$ .

$f (\frac{5 x}{3 (x - 2)}) = \frac{10 (x)}{x - 2} \cdot \frac{x - 2}{10}$

Step 5.3.8.2

Cancel the common factor.

$f (\frac{5 x}{3 (x - 2)}) = \frac{10 x}{x - 2} \cdot \frac{x - 2}{10}$

Step 5.3.8.3

Rewrite the expression.

$f (\frac{5 x}{3 (x - 2)}) = \frac{x}{x - 2} \cdot (x - 2)$

Step 5.3.9

Cancel the common factor of $x - 2$ .

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Step 5.3.9.1

Cancel the common factor.

$f (\frac{5 x}{3 (x - 2)}) = \frac{x}{x - 2} \cdot (x - 2)$

Step 5.3.9.2

Rewrite the expression.

$f (\frac{5 x}{3 (x - 2)}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{5 x}{3 (x - 2)}$ is the inverse of $f (x) = \frac{6 x}{3 x - 5}$ .

$f^{- 1} (x) = \frac{5 x}{3 (x - 2)}$