Precalculus Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply the equation by $y - 2$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Apply the distributive property.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $y - 2$ .

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Step 3.3.1.1

Cancel the common factor.

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

Step 3.3.1.2

Rewrite the expression.

$y x - 2 x = 3 y + 1$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $3 y$ from both sides of the equation.

$y x - 2 x - 3 y = 1$

Step 3.4.2

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

$y x - 3 y = 1 + 2 x$

Step 3.4.3

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

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Step 3.4.3.1

Factor $y$ out of $y x$ .

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

Step 3.4.3.2

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

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

Step 3.4.3.3

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

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

Step 3.4.4

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

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Step 3.4.4.1

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

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

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

Cancel the common factor of $x - 3$ .

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Step 3.4.4.2.1.1

Cancel the common factor.

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

Step 3.4.4.2.1.2

Divide $y$ by $1$ .

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

Step 3.4.4.3

Simplify the right side.

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Step 3.4.4.3.1

Combine the numerators over the common denominator.

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Multiply the numerator and denominator of the fraction by $x - 2$ .

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Step 5.2.3.1

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

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

Step 5.2.3.2

Combine.

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

Step 5.2.4

Apply the distributive property.

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

Step 5.2.5

Cancel the common factor of $x - 2$ .

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Step 5.2.5.1

Cancel the common factor.

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

Step 5.2.5.2

Rewrite the expression.

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

Step 5.2.6

Simplify the numerator.

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Step 5.2.6.1

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

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Step 5.2.6.1.1

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

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

Step 5.2.6.1.2

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

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

Step 5.2.6.2

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

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

Step 5.2.6.3

Write $1$ as a fraction with a common denominator.

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

Step 5.2.6.4

Combine the numerators over the common denominator.

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

Step 5.2.6.5

Reorder terms.

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

Step 5.2.6.6

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

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Step 5.2.6.6.1

Apply the distributive property.

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

Step 5.2.6.6.2

Multiply $3$ by $2$ .

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

Step 5.2.6.6.3

Multiply $2$ by $1$ .

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

Step 5.2.6.6.4

Add $x$ and $6 x$ .

$f^{- 1} (\frac{3 x + 1}{x - 2}) = \frac{(x - 2) (\frac{7 x + 2 - 2}{x - 2})}{3 x + 1 + (x - 2) \cdot - 3}$

Step 5.2.6.6.5

Subtract $2$ from $2$ .

$f^{- 1} (\frac{3 x + 1}{x - 2}) = \frac{(x - 2) (\frac{7 x + 0}{x - 2})}{3 x + 1 + (x - 2) \cdot - 3}$

Step 5.2.6.6.6

Add $7 x$ and $0$ .

$f^{- 1} (\frac{3 x + 1}{x - 2}) = \frac{(x - 2) (\frac{7 x}{x - 2})}{3 x + 1 + (x - 2) \cdot - 3}$

Step 5.2.7

Simplify the denominator.

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Step 5.2.7.1

Apply the distributive property.

$f^{- 1} (\frac{3 x + 1}{x - 2}) = \frac{(x - 2) (\frac{7 x}{x - 2})}{3 x + 1 + x \cdot - 3 - 2 \cdot - 3}$

Step 5.2.7.2

Move $- 3$ to the left of $x$ .

$f^{- 1} (\frac{3 x + 1}{x - 2}) = \frac{(x - 2) (\frac{7 x}{x - 2})}{3 x + 1 - 3 \cdot x - 2 \cdot - 3}$

Step 5.2.7.3

Multiply $- 2$ by $- 3$ .

$f^{- 1} (\frac{3 x + 1}{x - 2}) = \frac{(x - 2) (\frac{7 x}{x - 2})}{3 x + 1 - 3 \cdot x + 6}$

Step 5.2.7.4

Subtract $3 x$ from $3 x$ .

$f^{- 1} (\frac{3 x + 1}{x - 2}) = \frac{(x - 2) (\frac{7 x}{x - 2})}{0 + 1 + 6}$

Step 5.2.7.5

Add $0$ and $1$ .

$f^{- 1} (\frac{3 x + 1}{x - 2}) = \frac{(x - 2) (\frac{7 x}{x - 2})}{1 + 6}$

Step 5.2.7.6

Add $1$ and $6$ .

$f^{- 1} (\frac{3 x + 1}{x - 2}) = \frac{(x - 2) (\frac{7 x}{x - 2})}{7}$

Step 5.2.8

Reduce the expression by cancelling the common factors.

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Step 5.2.8.1

Factor $\frac{7 x}{x - 2}$ out of $\frac{(x - 2) \frac{7 x}{x - 2}}{7}$ .

$f^{- 1} (\frac{3 x + 1}{x - 2}) = \frac{7 x}{x - 2} \cdot \frac{x - 2}{7}$

Step 5.2.8.2

Cancel the common factor of $7$ .

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Step 5.2.8.2.1

Factor $7$ out of $7 x$ .

$f^{- 1} (\frac{3 x + 1}{x - 2}) = \frac{7 (x)}{x - 2} \cdot \frac{x - 2}{7}$

Step 5.2.8.2.2

Cancel the common factor.

$f^{- 1} (\frac{3 x + 1}{x - 2}) = \frac{7 x}{x - 2} \cdot \frac{x - 2}{7}$

Step 5.2.8.2.3

Rewrite the expression.

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

Step 5.2.8.3

Cancel the common factor of $x - 2$ .

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Step 5.2.8.3.1

Cancel the common factor.

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

Step 5.2.8.3.2

Rewrite the expression.

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

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

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

Step 5.3.3

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

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Step 5.3.3.1

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

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

Step 5.3.3.2

Combine.

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

Step 5.3.4

Apply the distributive property.

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

Step 5.3.5

Cancel the common factor of $x - 3$ .

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Step 5.3.5.1

Cancel the common factor.

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

Step 5.3.5.2

Rewrite the expression.

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

Step 5.3.6

Simplify the numerator.

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Step 5.3.6.1

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

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Step 5.3.6.1.1

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

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

Step 5.3.6.1.2

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

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

Step 5.3.6.1.3

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

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

Step 5.3.6.2

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

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

Step 5.3.6.3

Write $1$ as a fraction with a common denominator.

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

Step 5.3.6.4

Combine the numerators over the common denominator.

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

Step 5.3.6.5

Reorder terms.

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

Step 5.3.6.6

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

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Step 5.3.6.6.1

Apply the distributive property.

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

Step 5.3.6.6.2

Multiply $3$ by $1$ .

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

Step 5.3.6.6.3

Multiply $2$ by $3$ .

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

Step 5.3.6.6.4

Add $x$ and $6 x$ .

$f (\frac{1 + 2 x}{x - 3}) = \frac{(x - 3) (\frac{7 x + 3 - 3}{x - 3})}{1 + 2 x + (x - 3) \cdot - 2}$

Step 5.3.6.6.5

Subtract $3$ from $3$ .

$f (\frac{1 + 2 x}{x - 3}) = \frac{(x - 3) (\frac{7 x + 0}{x - 3})}{1 + 2 x + (x - 3) \cdot - 2}$

Step 5.3.6.6.6

Add $7 x$ and $0$ .

$f (\frac{1 + 2 x}{x - 3}) = \frac{(x - 3) (\frac{7 x}{x - 3})}{1 + 2 x + (x - 3) \cdot - 2}$

Step 5.3.7

Simplify the denominator.

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Step 5.3.7.1

Apply the distributive property.

$f (\frac{1 + 2 x}{x - 3}) = \frac{(x - 3) (\frac{7 x}{x - 3})}{1 + 2 x + x \cdot - 2 - 3 \cdot - 2}$

Step 5.3.7.2

Move $- 2$ to the left of $x$ .

$f (\frac{1 + 2 x}{x - 3}) = \frac{(x - 3) (\frac{7 x}{x - 3})}{1 + 2 x - 2 \cdot x - 3 \cdot - 2}$

Step 5.3.7.3

Multiply $- 3$ by $- 2$ .

$f (\frac{1 + 2 x}{x - 3}) = \frac{(x - 3) (\frac{7 x}{x - 3})}{1 + 2 x - 2 \cdot x + 6}$

Step 5.3.7.4

Add $1$ and $6$ .

$f (\frac{1 + 2 x}{x - 3}) = \frac{(x - 3) (\frac{7 x}{x - 3})}{2 x - 2 x + 7}$

Step 5.3.7.5

Subtract $2 x$ from $2 x$ .

$f (\frac{1 + 2 x}{x - 3}) = \frac{(x - 3) (\frac{7 x}{x - 3})}{0 + 7}$

Step 5.3.7.6

Add $0$ and $7$ .

$f (\frac{1 + 2 x}{x - 3}) = \frac{(x - 3) (\frac{7 x}{x - 3})}{7}$

Step 5.3.8

Reduce the expression by cancelling the common factors.

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Step 5.3.8.1

Factor $\frac{7 x}{x - 3}$ out of $\frac{(x - 3) \frac{7 x}{x - 3}}{7}$ .

$f (\frac{1 + 2 x}{x - 3}) = \frac{7 x}{x - 3} \cdot \frac{x - 3}{7}$

Step 5.3.8.2

Cancel the common factor of $7$ .

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Step 5.3.8.2.1

Factor $7$ out of $7 x$ .

$f (\frac{1 + 2 x}{x - 3}) = \frac{7 (x)}{x - 3} \cdot \frac{x - 3}{7}$

Step 5.3.8.2.2

Cancel the common factor.

$f (\frac{1 + 2 x}{x - 3}) = \frac{7 x}{x - 3} \cdot \frac{x - 3}{7}$

Step 5.3.8.2.3

Rewrite the expression.

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

Step 5.3.8.3

Cancel the common factor of $x - 3$ .

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Step 5.3.8.3.1

Cancel the common factor.

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

Step 5.3.8.3.2

Rewrite the expression.

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

Step 5.4

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

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