Precalculus Examples

$f (x) = \frac{8 x}{3 x - 7}$

Step 1

Write $f (x) = \frac{8 x}{3 x - 7}$ as an equation.

$y = \frac{8 x}{3 x - 7}$

Step 2

Interchange the variables.

$x = \frac{8 y}{3 y - 7}$

Step 3

Solve for $y$ .

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Step 3.1

Multiply the equation by $3 y - 7$ .

$(3 y - 7) (x) = (\frac{8 y}{3 y - 7}) (3 y - 7)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Apply the distributive property.

$3 y x - 7 x = (\frac{8 y}{3 y - 7}) (3 y - 7)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $3 y - 7$ .

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Step 3.3.1.1

Cancel the common factor.

$3 y x - 7 x = \frac{8 y}{3 y - 7} (3 y - 7)$

Step 3.3.1.2

Rewrite the expression.

$3 y x - 7 x = 8 y$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $8 y$ from both sides of the equation.

$3 y x - 7 x - 8 y = 0$

Step 3.4.2

Add $7 x$ to both sides of the equation.

$3 y x - 8 y = 7 x$

Step 3.4.3

Factor $y$ out of $3 y x - 8 y$ .

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Step 3.4.3.1

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

$y (3 x) - 8 y = 7 x$

Step 3.4.3.2

Factor $y$ out of $- 8 y$ .

$y (3 x) + y \cdot - 8 = 7 x$

Step 3.4.3.3

Factor $y$ out of $y (3 x) + y \cdot - 8$ .

$y (3 x - 8) = 7 x$

Step 3.4.4

Divide each term in $y (3 x - 8) = 7 x$ by $3 x - 8$ and simplify.

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Step 3.4.4.1

Divide each term in $y (3 x - 8) = 7 x$ by $3 x - 8$ .

$\frac{y (3 x - 8)}{3 x - 8} = \frac{7 x}{3 x - 8}$

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

Cancel the common factor of $3 x - 8$ .

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Step 3.4.4.2.1.1

Cancel the common factor.

$\frac{y (3 x - 8)}{3 x - 8} = \frac{7 x}{3 x - 8}$

Step 3.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{7 x}{3 x - 8}$

Step 4

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

$f^{- 1} (x) = \frac{7 x}{3 x - 8}$

Step 5

Verify if $f^{- 1} (x) = \frac{7 x}{3 x - 8}$ is the inverse of $f (x) = \frac{8 x}{3 x - 7}$ .

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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 $f^{- 1} (\frac{8 x}{3 x - 7})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{8 x}{3 x - 7}) = \frac{7 (\frac{8 x}{3 x - 7})}{3 (\frac{8 x}{3 x - 7}) - 8}$

Step 5.2.3

Combine $7$ and $\frac{8 x}{3 x - 7}$ .

$f^{- 1} (\frac{8 x}{3 x - 7}) = \frac{\frac{7 (8 x)}{3 x - 7}}{3 (\frac{8 x}{3 x - 7}) - 8}$

Step 5.2.4

Simplify the denominator.

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Step 5.2.4.1

Multiply $3 \frac{8 x}{3 x - 7}$ .

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Step 5.2.4.1.1

Combine $3$ and $\frac{8 x}{3 x - 7}$ .

$f^{- 1} (\frac{8 x}{3 x - 7}) = \frac{\frac{7 (8 x)}{3 x - 7}}{\frac{3 (8 x)}{3 x - 7} - 8}$

Step 5.2.4.1.2

Multiply $8$ by $3$ .

$f^{- 1} (\frac{8 x}{3 x - 7}) = \frac{\frac{7 (8 x)}{3 x - 7}}{\frac{24 x}{3 x - 7} - 8}$

Step 5.2.4.2

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{3 x - 7}{3 x - 7}$ .

$f^{- 1} (\frac{8 x}{3 x - 7}) = \frac{\frac{7 (8 x)}{3 x - 7}}{\frac{24 x}{3 x - 7} - 8 \cdot \frac{3 x - 7}{3 x - 7}}$

Step 5.2.4.3

Combine $- 8$ and $\frac{3 x - 7}{3 x - 7}$ .

$f^{- 1} (\frac{8 x}{3 x - 7}) = \frac{\frac{7 (8 x)}{3 x - 7}}{\frac{24 x}{3 x - 7} + \frac{- 8 (3 x - 7)}{3 x - 7}}$

Step 5.2.4.4

Combine the numerators over the common denominator.

$f^{- 1} (\frac{8 x}{3 x - 7}) = \frac{\frac{7 (8 x)}{3 x - 7}}{\frac{24 x - 8 (3 x - 7)}{3 x - 7}}$

Step 5.2.4.5

Rewrite $\frac{24 x - 8 (3 x - 7)}{3 x - 7}$ in a factored form.

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Step 5.2.4.5.1

Factor $8$ out of $24 x - 8 (3 x - 7)$ .

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Step 5.2.4.5.1.1

Factor $8$ out of $24 x$ .

$f^{- 1} (\frac{8 x}{3 x - 7}) = \frac{\frac{7 (8 x)}{3 x - 7}}{\frac{8 (3 x) - 8 (3 x - 7)}{3 x - 7}}$

Step 5.2.4.5.1.2

Factor $8$ out of $- 8 (3 x - 7)$ .

$f^{- 1} (\frac{8 x}{3 x - 7}) = \frac{\frac{7 (8 x)}{3 x - 7}}{\frac{8 (3 x) + 8 (- (3 x - 7))}{3 x - 7}}$

Step 5.2.4.5.1.3

Factor $8$ out of $8 (3 x) + 8 (- (3 x - 7))$ .

$f^{- 1} (\frac{8 x}{3 x - 7}) = \frac{\frac{7 (8 x)}{3 x - 7}}{\frac{8 (3 x - (3 x - 7))}{3 x - 7}}$

Step 5.2.4.5.2

Apply the distributive property.

$f^{- 1} (\frac{8 x}{3 x - 7}) = \frac{\frac{7 (8 x)}{3 x - 7}}{\frac{8 (3 x - (3 x) + 7)}{3 x - 7}}$

Step 5.2.4.5.3

Multiply $3$ by $- 1$ .

$f^{- 1} (\frac{8 x}{3 x - 7}) = \frac{\frac{7 (8 x)}{3 x - 7}}{\frac{8 (3 x - 3 x + 7)}{3 x - 7}}$

Step 5.2.4.5.4

Multiply $- 1$ by $- 7$ .

$f^{- 1} (\frac{8 x}{3 x - 7}) = \frac{\frac{7 (8 x)}{3 x - 7}}{\frac{8 (3 x - 3 x + 7)}{3 x - 7}}$

Step 5.2.4.5.5

Subtract $3 x$ from $3 x$ .

$f^{- 1} (\frac{8 x}{3 x - 7}) = \frac{\frac{7 (8 x)}{3 x - 7}}{\frac{8 (0 + 7)}{3 x - 7}}$

Step 5.2.4.5.6

Add $0$ and $7$ .

$f^{- 1} (\frac{8 x}{3 x - 7}) = \frac{\frac{7 (8 x)}{3 x - 7}}{\frac{8 \cdot 7}{3 x - 7}}$

Step 5.2.4.6

Multiply $8$ by $7$ .

$f^{- 1} (\frac{8 x}{3 x - 7}) = \frac{\frac{7 (8 x)}{3 x - 7}}{\frac{56}{3 x - 7}}$

Step 5.2.5

Multiply $7$ by $8$ .

$f^{- 1} (\frac{8 x}{3 x - 7}) = \frac{\frac{56 x}{3 x - 7}}{\frac{56}{3 x - 7}}$

Step 5.2.6

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{8 x}{3 x - 7}) = \frac{56 x}{3 x - 7} \cdot \frac{3 x - 7}{56}$

Step 5.2.7

Cancel the common factor of $56$ .

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Step 5.2.7.1

Factor $56$ out of $56 x$ .

$f^{- 1} (\frac{8 x}{3 x - 7}) = \frac{56 (x)}{3 x - 7} \cdot \frac{3 x - 7}{56}$

Step 5.2.7.2

Cancel the common factor.

$f^{- 1} (\frac{8 x}{3 x - 7}) = \frac{56 x}{3 x - 7} \cdot \frac{3 x - 7}{56}$

Step 5.2.7.3

Rewrite the expression.

$f^{- 1} (\frac{8 x}{3 x - 7}) = \frac{x}{3 x - 7} \cdot (3 x - 7)$

Step 5.2.8

Cancel the common factor of $3 x - 7$ .

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Step 5.2.8.1

Cancel the common factor.

$f^{- 1} (\frac{8 x}{3 x - 7}) = \frac{x}{3 x - 7} \cdot (3 x - 7)$

Step 5.2.8.2

Rewrite the expression.

$f^{- 1} (\frac{8 x}{3 x - 7}) = x$

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 $f (\frac{7 x}{3 x - 8})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{7 x}{3 x - 8}) = \frac{8 (\frac{7 x}{3 x - 8})}{3 (\frac{7 x}{3 x - 8}) - 7}$

Step 5.3.3

Combine $8$ and $\frac{7 x}{3 x - 8}$ .

$f (\frac{7 x}{3 x - 8}) = \frac{\frac{8 (7 x)}{3 x - 8}}{3 (\frac{7 x}{3 x - 8}) - 7}$

Step 5.3.4

Simplify the denominator.

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Step 5.3.4.1

Multiply $3 \frac{7 x}{3 x - 8}$ .

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Step 5.3.4.1.1

Combine $3$ and $\frac{7 x}{3 x - 8}$ .

$f (\frac{7 x}{3 x - 8}) = \frac{\frac{8 (7 x)}{3 x - 8}}{\frac{3 (7 x)}{3 x - 8} - 7}$

Step 5.3.4.1.2

Multiply $7$ by $3$ .

$f (\frac{7 x}{3 x - 8}) = \frac{\frac{8 (7 x)}{3 x - 8}}{\frac{21 x}{3 x - 8} - 7}$

Step 5.3.4.2

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{3 x - 8}{3 x - 8}$ .

$f (\frac{7 x}{3 x - 8}) = \frac{\frac{8 (7 x)}{3 x - 8}}{\frac{21 x}{3 x - 8} - 7 \cdot \frac{3 x - 8}{3 x - 8}}$

Step 5.3.4.3

Combine $- 7$ and $\frac{3 x - 8}{3 x - 8}$ .

$f (\frac{7 x}{3 x - 8}) = \frac{\frac{8 (7 x)}{3 x - 8}}{\frac{21 x}{3 x - 8} + \frac{- 7 (3 x - 8)}{3 x - 8}}$

Step 5.3.4.4

Combine the numerators over the common denominator.

$f (\frac{7 x}{3 x - 8}) = \frac{\frac{8 (7 x)}{3 x - 8}}{\frac{21 x - 7 (3 x - 8)}{3 x - 8}}$

Step 5.3.4.5

Rewrite $\frac{21 x - 7 (3 x - 8)}{3 x - 8}$ in a factored form.

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Step 5.3.4.5.1

Factor $7$ out of $21 x - 7 (3 x - 8)$ .

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Step 5.3.4.5.1.1

Factor $7$ out of $21 x$ .

$f (\frac{7 x}{3 x - 8}) = \frac{\frac{8 (7 x)}{3 x - 8}}{\frac{7 (3 x) - 7 (3 x - 8)}{3 x - 8}}$

Step 5.3.4.5.1.2

Factor $7$ out of $- 7 (3 x - 8)$ .

$f (\frac{7 x}{3 x - 8}) = \frac{\frac{8 (7 x)}{3 x - 8}}{\frac{7 (3 x) + 7 (- (3 x - 8))}{3 x - 8}}$

Step 5.3.4.5.1.3

Factor $7$ out of $7 (3 x) + 7 (- (3 x - 8))$ .

$f (\frac{7 x}{3 x - 8}) = \frac{\frac{8 (7 x)}{3 x - 8}}{\frac{7 (3 x - (3 x - 8))}{3 x - 8}}$

Step 5.3.4.5.2

Apply the distributive property.

$f (\frac{7 x}{3 x - 8}) = \frac{\frac{8 (7 x)}{3 x - 8}}{\frac{7 (3 x - (3 x) + 8)}{3 x - 8}}$

Step 5.3.4.5.3

Multiply $3$ by $- 1$ .

$f (\frac{7 x}{3 x - 8}) = \frac{\frac{8 (7 x)}{3 x - 8}}{\frac{7 (3 x - 3 x + 8)}{3 x - 8}}$

Step 5.3.4.5.4

Multiply $- 1$ by $- 8$ .

$f (\frac{7 x}{3 x - 8}) = \frac{\frac{8 (7 x)}{3 x - 8}}{\frac{7 (3 x - 3 x + 8)}{3 x - 8}}$

Step 5.3.4.5.5

Subtract $3 x$ from $3 x$ .

$f (\frac{7 x}{3 x - 8}) = \frac{\frac{8 (7 x)}{3 x - 8}}{\frac{7 (0 + 8)}{3 x - 8}}$

Step 5.3.4.5.6

Add $0$ and $8$ .

$f (\frac{7 x}{3 x - 8}) = \frac{\frac{8 (7 x)}{3 x - 8}}{\frac{7 \cdot 8}{3 x - 8}}$

Step 5.3.4.6

Multiply $7$ by $8$ .

$f (\frac{7 x}{3 x - 8}) = \frac{\frac{8 (7 x)}{3 x - 8}}{\frac{56}{3 x - 8}}$

Step 5.3.5

Multiply $8$ by $7$ .

$f (\frac{7 x}{3 x - 8}) = \frac{\frac{56 x}{3 x - 8}}{\frac{56}{3 x - 8}}$

Step 5.3.6

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{7 x}{3 x - 8}) = \frac{56 x}{3 x - 8} \cdot \frac{3 x - 8}{56}$

Step 5.3.7

Cancel the common factor of $56$ .

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Step 5.3.7.1

Factor $56$ out of $56 x$ .

$f (\frac{7 x}{3 x - 8}) = \frac{56 (x)}{3 x - 8} \cdot \frac{3 x - 8}{56}$

Step 5.3.7.2

Cancel the common factor.

$f (\frac{7 x}{3 x - 8}) = \frac{56 x}{3 x - 8} \cdot \frac{3 x - 8}{56}$

Step 5.3.7.3

Rewrite the expression.

$f (\frac{7 x}{3 x - 8}) = \frac{x}{3 x - 8} \cdot (3 x - 8)$

Step 5.3.8

Cancel the common factor of $3 x - 8$ .

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Step 5.3.8.1

Cancel the common factor.

$f (\frac{7 x}{3 x - 8}) = \frac{x}{3 x - 8} \cdot (3 x - 8)$

Step 5.3.8.2

Rewrite the expression.

$f (\frac{7 x}{3 x - 8}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{7 x}{3 x - 8}$ is the inverse of $f (x) = \frac{8 x}{3 x - 7}$ .

$f^{- 1} (x) = \frac{7 x}{3 x - 8}$