Precalculus Examples

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

Step 1

Write $f (x) = \frac{2 x}{3 x - 1}$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply the equation by $3 y - 1$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify $(3 y - 1) (x)$ .

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Step 3.2.1.1

Apply the distributive property.

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

Step 3.2.1.2

Rewrite $- 1 x$ as $- x$ .

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

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

Cancel the common factor.

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

Step 3.3.1.2

Rewrite the expression.

$3 y x - x = 2 y$

Step 3.4

Solve for $y$ .

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Step 3.4.1

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

$3 y x - x - 2 y = 0$

Step 3.4.2

Add $x$ to both sides of the equation.

$3 y x - 2 y = x$

Step 3.4.3

Factor $y$ out of $3 y x - 2 y$ .

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Step 3.4.3.1

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

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

Step 3.4.3.2

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

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

Step 3.4.3.3

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

$y (3 x - 2) = x$

Step 3.4.4

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

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Step 3.4.4.1

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

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

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

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

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Step 3.4.4.2.1.1

Cancel the common factor.

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

Step 3.4.4.2.1.2

Divide $y$ by $1$ .

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2.4

Simplify the denominator.

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Step 5.2.4.1

Multiply $3 \frac{2 x}{3 x - 1}$ .

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Step 5.2.4.1.1

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

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

Step 5.2.4.1.2

Multiply $2$ by $3$ .

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

Step 5.2.4.2

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

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

Step 5.2.4.3

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

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

Step 5.2.4.4

Combine the numerators over the common denominator.

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

Step 5.2.4.5

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

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Step 5.2.4.5.1

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

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Step 5.2.4.5.1.1

Factor $2$ out of $6 x$ .

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

Step 5.2.4.5.1.2

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

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

Step 5.2.4.5.1.3

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

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

Step 5.2.4.5.2

Apply the distributive property.

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

Step 5.2.4.5.3

Multiply $3$ by $- 1$ .

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

Step 5.2.4.5.4

Multiply $- 1$ by $- 1$ .

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

Step 5.2.4.5.5

Subtract $3 x$ from $3 x$ .

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

Step 5.2.4.5.6

Add $0$ and $1$ .

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

Step 5.2.4.6

Multiply $2$ by $1$ .

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

Step 5.2.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2.6

Multiply $\frac{3 x - 1}{2}$ by $1$ .

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

Step 5.2.7

Cancel the common factor of $2$ .

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Step 5.2.7.1

Factor $2$ out of $2 x$ .

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

Step 5.2.7.2

Cancel the common factor.

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

Step 5.2.7.3

Rewrite the expression.

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

Step 5.2.8

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

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Step 5.2.8.1

Cancel the common factor.

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

Step 5.2.8.2

Rewrite the expression.

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

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

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

Step 5.3.3

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

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

Step 5.3.4

Simplify the denominator.

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Step 5.3.4.1

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

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

Step 5.3.4.2

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

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

Step 5.3.4.3

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

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

Step 5.3.4.4

Combine the numerators over the common denominator.

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

Step 5.3.4.5

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

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Step 5.3.4.5.1

Apply the distributive property.

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

Step 5.3.4.5.2

Multiply $3$ by $- 1$ .

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

Step 5.3.4.5.3

Multiply $- 1$ by $- 2$ .

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

Step 5.3.4.5.4

Subtract $3 x$ from $3 x$ .

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

Step 5.3.4.5.5

Add $0$ and $2$ .

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

Step 5.3.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.6

Cancel the common factor of $2$ .

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Step 5.3.6.1

Factor $2$ out of $2 x$ .

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

Step 5.3.6.2

Cancel the common factor.

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

Step 5.3.6.3

Rewrite the expression.

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

Step 5.3.7

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

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Step 5.3.7.1

Cancel the common factor.

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

Step 5.3.7.2

Rewrite the expression.

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

Step 5.4

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

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