Algebra Examples

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

Step 1

Write $f (x) = \frac{7 - 2 x}{3 x}$ as an equation.

$y = \frac{7 - 2 x}{3 x}$

Step 2

Interchange the variables.

$x = \frac{7 - 2 y}{3 y}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{7 - 2 y}{3 y} = x$ .

$\frac{7 - 2 y}{3 y} = x$

Step 3.2

Find the LCD of the terms in the equation.

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Step 3.2.1

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

$3 y, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$3 y$

Step 3.3

Multiply each term in $\frac{7 - 2 y}{3 y} = x$ by $3 y$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{7 - 2 y}{3 y} = x$ by $3 y$ .

$\frac{7 - 2 y}{3 y} (3 y) = x (3 y)$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Rewrite using the commutative property of multiplication.

$3 \frac{7 - 2 y}{3 y} y = x (3 y)$

Step 3.3.2.2

Cancel the common factor of $3$ .

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Step 3.3.2.2.1

Factor $3$ out of $3 y$ .

$3 \frac{7 - 2 y}{3 (y)} y = x (3 y)$

Step 3.3.2.2.2

Cancel the common factor.

$3 \frac{7 - 2 y}{3 y} y = x (3 y)$

Step 3.3.2.2.3

Rewrite the expression.

$\frac{7 - 2 y}{y} y = x (3 y)$

Step 3.3.2.3

Cancel the common factor of $y$ .

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Step 3.3.2.3.1

Cancel the common factor.

$\frac{7 - 2 y}{y} y = x (3 y)$

Step 3.3.2.3.2

Rewrite the expression.

$7 - 2 y = x (3 y)$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Rewrite using the commutative property of multiplication.

$7 - 2 y = 3 x y$

Step 3.4

Solve the equation.

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Step 3.4.1

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

$7 - 2 y - 3 x y = 0$

Step 3.4.2

Subtract $7$ from both sides of the equation.

$- 2 y - 3 x y = - 7$

Step 3.4.3

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

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Step 3.4.3.1

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

$y \cdot - 2 - 3 x y = - 7$

Step 3.4.3.2

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

$y \cdot - 2 + y (- 3 x) = - 7$

Step 3.4.3.3

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

$y (- 2 - 3 x) = - 7$

Step 3.4.4

Divide each term in $y (- 2 - 3 x) = - 7$ by $- 2 - 3 x$ and simplify.

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Step 3.4.4.1

Divide each term in $y (- 2 - 3 x) = - 7$ by $- 2 - 3 x$ .

$\frac{y (- 2 - 3 x)}{- 2 - 3 x} = \frac{- 7}{- 2 - 3 x}$

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

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

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Step 3.4.4.2.1.1

Cancel the common factor.

$\frac{y (- 2 - 3 x)}{- 2 - 3 x} = \frac{- 7}{- 2 - 3 x}$

Step 3.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 7}{- 2 - 3 x}$

Step 3.4.4.3

Simplify the right side.

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Step 3.4.4.3.1

Move the negative in front of the fraction.

$y = - \frac{7}{- 2 - 3 x}$

Step 3.4.4.3.2

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 3.4.4.3.3

Factor $- 1$ out of $- 3 x$ .

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

Step 3.4.4.3.4

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

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

Step 3.4.4.3.5

Simplify the expression.

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Step 3.4.4.3.5.1

Move the negative in front of the fraction.

$y = - - \frac{7}{2 + 3 x}$

Step 3.4.4.3.5.2

Multiply $- 1$ by $- 1$ .

$y = 1 \frac{7}{2 + 3 x}$

Step 3.4.4.3.5.3

Multiply $\frac{7}{2 + 3 x}$ by $1$ .

$y = \frac{7}{2 + 3 x}$

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = \frac{7}{2 + 3 x}$ is the inverse of $f (x) = \frac{7 - 2 x}{3 x}$ .

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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 $f^{- 1} (\frac{7 - 2 x}{3 x})$ by substituting in the value of $f$ into $f^{- 1}$ .

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

Step 5.2.3

Simplify the denominator.

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Step 5.2.3.1

Cancel the common factor of $3$ .

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Step 5.2.3.1.1

Factor $3$ out of $3 x$ .

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

Step 5.2.3.1.2

Cancel the common factor.

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

Step 5.2.3.1.3

Rewrite the expression.

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

Step 5.2.3.2

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

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

Step 5.2.3.3

Combine the numerators over the common denominator.

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

Step 5.2.3.4

Reorder terms.

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

Step 5.2.3.5

Rewrite $\frac{2 x - 2 x + 7}{x}$ in a factored form.

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Step 5.2.3.5.1

Subtract $2 x$ from $2 x$ .

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

Step 5.2.3.5.2

Add $0$ and $7$ .

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

Step 5.2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2.5

Cancel the common factor of $7$ .

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Step 5.2.5.1

Cancel the common factor.

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

Step 5.2.5.2

Rewrite the expression.

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

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 $f (\frac{7}{2 + 3 x})$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Multiply $- 2 \frac{7}{2 + 3 x}$ .

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Step 5.3.3.1.1

Combine $- 2$ and $\frac{7}{2 + 3 x}$ .

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

Step 5.3.3.1.2

Multiply $- 2$ by $7$ .

$f (\frac{7}{2 + 3 x}) = \frac{7 + \frac{- 14}{2 + 3 x}}{3 (\frac{7}{2 + 3 x})}$

Step 5.3.3.2

Move the negative in front of the fraction.

$f (\frac{7}{2 + 3 x}) = \frac{7 - \frac{14}{2 + 3 x}}{3 (\frac{7}{2 + 3 x})}$

Step 5.3.3.3

To write $7$ as a fraction with a common denominator, multiply by $\frac{2 + 3 x}{2 + 3 x}$ .

$f (\frac{7}{2 + 3 x}) = \frac{\frac{7 (2 + 3 x)}{2 + 3 x} - \frac{14}{2 + 3 x}}{3 (\frac{7}{2 + 3 x})}$

Step 5.3.3.4

Combine the numerators over the common denominator.

$f (\frac{7}{2 + 3 x}) = \frac{\frac{7 (2 + 3 x) - 14}{2 + 3 x}}{3 (\frac{7}{2 + 3 x})}$

Step 5.3.3.5

Reorder terms.

$f (\frac{7}{2 + 3 x}) = \frac{\frac{7 (2 + 3 x) - 14}{3 x + 2}}{3 (\frac{7}{2 + 3 x})}$

Step 5.3.3.6

Rewrite $\frac{7 (2 + 3 x) - 14}{3 x + 2}$ in a factored form.

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Step 5.3.3.6.1

Factor $7$ out of $7 (2 + 3 x) - 14$ .

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Step 5.3.3.6.1.1

Factor $7$ out of $- 14$ .

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

Step 5.3.3.6.1.2

Factor $7$ out of $7 (2 + 3 x) + 7 \cdot - 2$ .

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

Step 5.3.3.6.2

Subtract $2$ from $2$ .

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

Step 5.3.3.6.3

Add $3 x$ and $0$ .

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

Step 5.3.3.6.4

Multiply $7$ by $3$ .

$f (\frac{7}{2 + 3 x}) = \frac{\frac{21 x}{3 x + 2}}{3 (\frac{7}{2 + 3 x})}$

Step 5.3.4

Combine fractions.

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Step 5.3.4.1

Combine $3$ and $\frac{7}{2 + 3 x}$ .

$f (\frac{7}{2 + 3 x}) = \frac{\frac{21 x}{3 x + 2}}{\frac{3 \cdot 7}{2 + 3 x}}$

Step 5.3.4.2

Multiply $3$ by $7$ .

$f (\frac{7}{2 + 3 x}) = \frac{\frac{21 x}{3 x + 2}}{\frac{21}{2 + 3 x}}$

Step 5.3.5

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{7}{2 + 3 x}) = \frac{21 x}{3 x + 2} \cdot \frac{2 + 3 x}{21}$

Step 5.3.6

Cancel the common factor of $21$ .

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Step 5.3.6.1

Factor $21$ out of $21 x$ .

$f (\frac{7}{2 + 3 x}) = \frac{21 (x)}{3 x + 2} \cdot \frac{2 + 3 x}{21}$

Step 5.3.6.2

Cancel the common factor.

$f (\frac{7}{2 + 3 x}) = \frac{21 x}{3 x + 2} \cdot \frac{2 + 3 x}{21}$

Step 5.3.6.3

Rewrite the expression.

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

Step 5.3.7

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

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

Step 5.3.8

Cancel the common factor of $2 + 3 x$ and $3 x + 2$ .

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Step 5.3.8.1

Reorder terms.

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

Step 5.3.8.2

Cancel the common factor.

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

Step 5.3.8.3

Divide $x$ by $1$ .

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

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{7}{2 + 3 x}$ is the inverse of $f (x) = \frac{7 - 2 x}{3 x}$ .

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