Precalculus Examples

$y = \frac{6}{9 x - 1}$

Step 1

Interchange the variables.

$x = \frac{6}{9 y - 1}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{6}{9 y - 1} = x$ .

$\frac{6}{9 y - 1} = x$

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$9 y - 1, 1$

Step 2.2.2

Remove parentheses.

$9 y - 1, 1$

Step 2.2.3

The LCM of one and any expression is the expression.

$9 y - 1$

Step 2.3

Multiply each term in $\frac{6}{9 y - 1} = x$ by $9 y - 1$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $\frac{6}{9 y - 1} = x$ by $9 y - 1$ .

$\frac{6}{9 y - 1} (9 y - 1) = x (9 y - 1)$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $9 y - 1$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{6}{9 y - 1} (9 y - 1) = x (9 y - 1)$

Step 2.3.2.1.2

Rewrite the expression.

$6 = x (9 y - 1)$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Simplify by multiplying through.

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Step 2.3.3.1.1

Apply the distributive property.

$6 = x (9 y) + x \cdot - 1$

Step 2.3.3.1.2

Reorder.

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Step 2.3.3.1.2.1

Rewrite using the commutative property of multiplication.

$6 = 9 x y + x \cdot - 1$

Step 2.3.3.1.2.2

Move $- 1$ to the left of $x$ .

$6 = 9 x y - 1 \cdot x$

Step 2.3.3.2

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

$6 = 9 x y - x$

Step 2.4

Solve the equation.

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Step 2.4.1

Rewrite the equation as $9 x y - x = 6$ .

$9 x y - x = 6$

Step 2.4.2

Add $x$ to both sides of the equation.

$9 x y = 6 + x$

Step 2.4.3

Divide each term in $9 x y = 6 + x$ by $9 x$ and simplify.

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Step 2.4.3.1

Divide each term in $9 x y = 6 + x$ by $9 x$ .

$\frac{9 x y}{9 x} = \frac{6}{9 x} + \frac{x}{9 x}$

Step 2.4.3.2

Simplify the left side.

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Step 2.4.3.2.1

Cancel the common factor of $9$ .

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Step 2.4.3.2.1.1

Cancel the common factor.

$\frac{9 x y}{9 x} = \frac{6}{9 x} + \frac{x}{9 x}$

Step 2.4.3.2.1.2

Rewrite the expression.

$\frac{x y}{x} = \frac{6}{9 x} + \frac{x}{9 x}$

Step 2.4.3.2.2

Cancel the common factor of $x$ .

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Step 2.4.3.2.2.1

Cancel the common factor.

$\frac{x y}{x} = \frac{6}{9 x} + \frac{x}{9 x}$

Step 2.4.3.2.2.2

Divide $y$ by $1$ .

$y = \frac{6}{9 x} + \frac{x}{9 x}$

Step 2.4.3.3

Simplify the right side.

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Step 2.4.3.3.1

Simplify each term.

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Step 2.4.3.3.1.1

Cancel the common factor of $6$ and $9$ .

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Step 2.4.3.3.1.1.1

Factor $3$ out of $6$ .

$y = \frac{3 (2)}{9 x} + \frac{x}{9 x}$

Step 2.4.3.3.1.1.2

Cancel the common factors.

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Step 2.4.3.3.1.1.2.1

Factor $3$ out of $9 x$ .

$y = \frac{3 (2)}{3 (3 x)} + \frac{x}{9 x}$

Step 2.4.3.3.1.1.2.2

Cancel the common factor.

$y = \frac{3 \cdot 2}{3 (3 x)} + \frac{x}{9 x}$

Step 2.4.3.3.1.1.2.3

Rewrite the expression.

$y = \frac{2}{3 x} + \frac{x}{9 x}$

Step 2.4.3.3.1.2

Cancel the common factor of $x$ .

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Step 2.4.3.3.1.2.1

Cancel the common factor.

$y = \frac{2}{3 x} + \frac{x}{9 x}$

Step 2.4.3.3.1.2.2

Rewrite the expression.

$y = \frac{2}{3 x} + \frac{1}{9}$

Step 3

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

$f^{- 1} (x) = \frac{2}{3 x} + \frac{1}{9}$

Step 4

Verify if $f^{- 1} (x) = \frac{2}{3 x} + \frac{1}{9}$ is the inverse of $f (x) = \frac{6}{9 x - 1}$ .

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Step 4.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 4.2

Evaluate $f^{- 1} (f (x))$ .

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Step 4.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 4.2.2

Evaluate $f^{- 1} (\frac{6}{9 x - 1})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{6}{9 x - 1}) = \frac{2}{3 (\frac{6}{9 x - 1})} + \frac{1}{9}$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Combine $3$ and $\frac{6}{9 x - 1}$ .

$f^{- 1} (\frac{6}{9 x - 1}) = \frac{2}{\frac{3 \cdot 6}{9 x - 1}} + \frac{1}{9}$

Step 4.2.3.2

Multiply $3$ by $6$ .

$f^{- 1} (\frac{6}{9 x - 1}) = \frac{2}{\frac{18}{9 x - 1}} + \frac{1}{9}$

Step 4.2.3.3

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{6}{9 x - 1}) = 2 (\frac{9 x - 1}{18}) + \frac{1}{9}$

Step 4.2.3.4

Cancel the common factor of $2$ .

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Step 4.2.3.4.1

Factor $2$ out of $18$ .

$f^{- 1} (\frac{6}{9 x - 1}) = 2 (\frac{9 x - 1}{2 (9)}) + \frac{1}{9}$

Step 4.2.3.4.2

Cancel the common factor.

$f^{- 1} (\frac{6}{9 x - 1}) = 2 (\frac{9 x - 1}{2 \cdot 9}) + \frac{1}{9}$

Step 4.2.3.4.3

Rewrite the expression.

$f^{- 1} (\frac{6}{9 x - 1}) = \frac{9 x - 1}{9} + \frac{1}{9}$

Step 4.2.4

Simplify terms.

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Step 4.2.4.1

Combine the numerators over the common denominator.

$f^{- 1} (\frac{6}{9 x - 1}) = \frac{9 x - 1 + 1}{9}$

Step 4.2.4.2

Combine the opposite terms in $9 x - 1 + 1$ .

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Step 4.2.4.2.1

Add $- 1$ and $1$ .

$f^{- 1} (\frac{6}{9 x - 1}) = \frac{9 x + 0}{9}$

Step 4.2.4.2.2

Add $9 x$ and $0$ .

$f^{- 1} (\frac{6}{9 x - 1}) = \frac{9 x}{9}$

Step 4.2.4.3

Cancel the common factor of $9$ .

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Step 4.2.4.3.1

Cancel the common factor.

$f^{- 1} (\frac{6}{9 x - 1}) = \frac{9 x}{9}$

Step 4.2.4.3.2

Divide $x$ by $1$ .

$f^{- 1} (\frac{6}{9 x - 1}) = x$

Step 4.3

Evaluate $f (f^{- 1} (x))$ .

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Step 4.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 4.3.2

Evaluate $f (\frac{2}{3 x} + \frac{1}{9})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{2}{3 x} + \frac{1}{9}) = \frac{6}{9 (\frac{2}{3 x} + \frac{1}{9}) - 1}$

Step 4.3.3

Simplify the denominator.

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Step 4.3.3.1

Apply the distributive property.

$f (\frac{2}{3 x} + \frac{1}{9}) = \frac{6}{9 (\frac{2}{3 x}) + 9 (\frac{1}{9}) - 1}$

Step 4.3.3.2

Cancel the common factor of $3$ .

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Step 4.3.3.2.1

Factor $3$ out of $9$ .

$f (\frac{2}{3 x} + \frac{1}{9}) = \frac{6}{3 (3) (\frac{2}{3 x}) + 9 (\frac{1}{9}) - 1}$

Step 4.3.3.2.2

Factor $3$ out of $3 x$ .

$f (\frac{2}{3 x} + \frac{1}{9}) = \frac{6}{3 (3) (\frac{2}{3 (x)}) + 9 (\frac{1}{9}) - 1}$

Step 4.3.3.2.3

Cancel the common factor.

$f (\frac{2}{3 x} + \frac{1}{9}) = \frac{6}{3 \cdot (3 (\frac{2}{3 x})) + 9 (\frac{1}{9}) - 1}$

Step 4.3.3.2.4

Rewrite the expression.

$f (\frac{2}{3 x} + \frac{1}{9}) = \frac{6}{3 (\frac{2}{x}) + 9 (\frac{1}{9}) - 1}$

Step 4.3.3.3

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

$f (\frac{2}{3 x} + \frac{1}{9}) = \frac{6}{\frac{3 \cdot 2}{x} + 9 (\frac{1}{9}) - 1}$

Step 4.3.3.4

Multiply $3$ by $2$ .

$f (\frac{2}{3 x} + \frac{1}{9}) = \frac{6}{\frac{6}{x} + 9 (\frac{1}{9}) - 1}$

Step 4.3.3.5

Cancel the common factor of $9$ .

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Step 4.3.3.5.1

Cancel the common factor.

$f (\frac{2}{3 x} + \frac{1}{9}) = \frac{6}{\frac{6}{x} + 9 (\frac{1}{9}) - 1}$

Step 4.3.3.5.2

Rewrite the expression.

$f (\frac{2}{3 x} + \frac{1}{9}) = \frac{6}{\frac{6}{x} + 1 - 1}$

Step 4.3.3.6

Subtract $1$ from $1$ .

$f (\frac{2}{3 x} + \frac{1}{9}) = \frac{6}{\frac{6}{x} + 0}$

Step 4.3.3.7

Add $\frac{6}{x}$ and $0$ .

$f (\frac{2}{3 x} + \frac{1}{9}) = \frac{6}{\frac{6}{x}}$

Step 4.3.4

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{2}{3 x} + \frac{1}{9}) = 6 (\frac{x}{6})$

Step 4.3.5

Cancel the common factor of $6$ .

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Step 4.3.5.1

Cancel the common factor.

$f (\frac{2}{3 x} + \frac{1}{9}) = 6 (\frac{x}{6})$

Step 4.3.5.2

Rewrite the expression.

$f (\frac{2}{3 x} + \frac{1}{9}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{2}{3 x} + \frac{1}{9}$ is the inverse of $f (x) = \frac{6}{9 x - 1}$ .

$f^{- 1} (x) = \frac{2}{3 x} + \frac{1}{9}$