Finite Math Examples

$f (x) = \frac{7 x + 4}{x - 5}$

Step 1

Write $f (x) = \frac{7 x + 4}{x - 5}$ as an equation.

$y = \frac{7 x + 4}{x - 5}$

Step 2

Interchange the variables.

$x = \frac{7 y + 4}{y - 5}$

Step 3

Solve for $y$ .

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Step 3.1

Multiply the equation by $y - 5$ .

$(y - 5) (x) = (\frac{7 y + 4}{y - 5}) (y - 5)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Apply the distributive property.

$y x - 5 x = (\frac{7 y + 4}{y - 5}) (y - 5)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $y - 5$ .

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Step 3.3.1.1

Cancel the common factor.

$y x - 5 x = \frac{7 y + 4}{y - 5} (y - 5)$

Step 3.3.1.2

Rewrite the expression.

$y x - 5 x = 7 y + 4$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $7 y$ from both sides of the equation.

$y x - 5 x - 7 y = 4$

Step 3.4.2

Add $5 x$ to both sides of the equation.

$y x - 7 y = 4 + 5 x$

Step 3.4.3

Factor $y$ out of $y x - 7 y$ .

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Step 3.4.3.1

Factor $y$ out of $y x$ .

$y (x) - 7 y = 4 + 5 x$

Step 3.4.3.2

Factor $y$ out of $- 7 y$ .

$y (x) + y \cdot - 7 = 4 + 5 x$

Step 3.4.3.3

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

$y (x - 7) = 4 + 5 x$

Step 3.4.4

Divide each term in $y (x - 7) = 4 + 5 x$ by $x - 7$ and simplify.

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Step 3.4.4.1

Divide each term in $y (x - 7) = 4 + 5 x$ by $x - 7$ .

$\frac{y (x - 7)}{x - 7} = \frac{4}{x - 7} + \frac{5 x}{x - 7}$

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

Cancel the common factor of $x - 7$ .

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Step 3.4.4.2.1.1

Cancel the common factor.

$\frac{y (x - 7)}{x - 7} = \frac{4}{x - 7} + \frac{5 x}{x - 7}$

Step 3.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{4}{x - 7} + \frac{5 x}{x - 7}$

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.

$y = \frac{4 + 5 x}{x - 7}$

Step 4

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

$f^{- 1} (x) = \frac{4 + 5 x}{x - 7}$

Step 5

Verify if $f^{- 1} (x) = \frac{4 + 5 x}{x - 7}$ is the inverse of $f (x) = \frac{7 x + 4}{x - 5}$ .

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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 $f^{- 1} (\frac{7 x + 4}{x - 5})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{4 + 5 (\frac{7 x + 4}{x - 5})}{(\frac{7 x + 4}{x - 5}) - 7}$

Step 5.2.3

Multiply the numerator and denominator of the fraction by $x - 5$ .

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Step 5.2.3.1

Multiply $\frac{4 + 5 \frac{7 x + 4}{x - 5}}{\frac{7 x + 4}{x - 5} - 7}$ by $\frac{x - 5}{x - 5}$ .

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{x - 5}{x - 5} \cdot \frac{4 + 5 (\frac{7 x + 4}{x - 5})}{\frac{7 x + 4}{x - 5} - 7}$

Step 5.2.3.2

Combine.

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{(x - 5) (4 + 5 (\frac{7 x + 4}{x - 5}))}{(x - 5) (\frac{7 x + 4}{x - 5} - 7)}$

Step 5.2.4

Apply the distributive property.

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{(x - 5) \cdot 4 + (x - 5) (5 (\frac{7 x + 4}{x - 5}))}{(x - 5) (\frac{7 x + 4}{x - 5}) + (x - 5) \cdot - 7}$

Step 5.2.5

Cancel the common factor of $x - 5$ .

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Step 5.2.5.1

Cancel the common factor.

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{(x - 5) \cdot 4 + (x - 5) (5 (\frac{7 x + 4}{x - 5}))}{(x - 5) (\frac{7 x + 4}{x - 5}) + (x - 5) \cdot - 7}$

Step 5.2.5.2

Rewrite the expression.

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{(x - 5) \cdot 4 + (x - 5) (5 (\frac{7 x + 4}{x - 5}))}{7 x + 4 + (x - 5) \cdot - 7}$

Step 5.2.6

Simplify the numerator.

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Step 5.2.6.1

Factor $x - 5$ out of $(x - 5) \cdot 4 + (x - 5) \cdot 5 \frac{7 x + 4}{x - 5}$ .

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Step 5.2.6.1.1

Factor $x - 5$ out of $(x - 5) \cdot 4$ .

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{(x - 5) (4) + (x - 5) (5 (\frac{7 x + 4}{x - 5}))}{7 x + 4 + (x - 5) \cdot - 7}$

Step 5.2.6.1.2

Factor $x - 5$ out of $(x - 5) (4) + (x - 5) (5 \frac{7 x + 4}{x - 5})$ .

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{(x - 5) (4 + 5 (\frac{7 x + 4}{x - 5}))}{7 x + 4 + (x - 5) \cdot - 7}$

Step 5.2.6.2

Combine $5$ and $\frac{7 x + 4}{x - 5}$ .

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{(x - 5) (4 + \frac{5 (7 x + 4)}{x - 5})}{7 x + 4 + (x - 5) \cdot - 7}$

Step 5.2.6.3

To write $4$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{(x - 5) (\frac{4 (x - 5)}{x - 5} + \frac{5 (7 x + 4)}{x - 5})}{7 x + 4 + (x - 5) \cdot - 7}$

Step 5.2.6.4

Combine the numerators over the common denominator.

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{(x - 5) (\frac{4 (x - 5) + 5 (7 x + 4)}{x - 5})}{7 x + 4 + (x - 5) \cdot - 7}$

Step 5.2.6.5

Reorder terms.

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{(x - 5) (\frac{5 (7 x + 4) + 4 (x - 5)}{x - 5})}{7 x + 4 + (x - 5) \cdot - 7}$

Step 5.2.6.6

Rewrite $\frac{5 (7 x + 4) + 4 (x - 5)}{x - 5}$ in a factored form.

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Step 5.2.6.6.1

Apply the distributive property.

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{(x - 5) (\frac{5 (7 x) + 5 \cdot 4 + 4 (x - 5)}{x - 5})}{7 x + 4 + (x - 5) \cdot - 7}$

Step 5.2.6.6.2

Multiply $7$ by $5$ .

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{(x - 5) (\frac{35 x + 5 \cdot 4 + 4 (x - 5)}{x - 5})}{7 x + 4 + (x - 5) \cdot - 7}$

Step 5.2.6.6.3

Multiply $5$ by $4$ .

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{(x - 5) (\frac{35 x + 20 + 4 (x - 5)}{x - 5})}{7 x + 4 + (x - 5) \cdot - 7}$

Step 5.2.6.6.4

Apply the distributive property.

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{(x - 5) (\frac{35 x + 20 + 4 x + 4 \cdot - 5}{x - 5})}{7 x + 4 + (x - 5) \cdot - 7}$

Step 5.2.6.6.5

Multiply $4$ by $- 5$ .

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{(x - 5) (\frac{35 x + 20 + 4 x - 20}{x - 5})}{7 x + 4 + (x - 5) \cdot - 7}$

Step 5.2.6.6.6

Add $35 x$ and $4 x$ .

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{(x - 5) (\frac{39 x + 20 - 20}{x - 5})}{7 x + 4 + (x - 5) \cdot - 7}$

Step 5.2.6.6.7

Subtract $20$ from $20$ .

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{(x - 5) (\frac{39 x + 0}{x - 5})}{7 x + 4 + (x - 5) \cdot - 7}$

Step 5.2.6.6.8

Add $39 x$ and $0$ .

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{(x - 5) (\frac{39 x}{x - 5})}{7 x + 4 + (x - 5) \cdot - 7}$

Step 5.2.7

Simplify the denominator.

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Step 5.2.7.1

Apply the distributive property.

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{(x - 5) (\frac{39 x}{x - 5})}{7 x + 4 + x \cdot - 7 - 5 \cdot - 7}$

Step 5.2.7.2

Move $- 7$ to the left of $x$ .

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{(x - 5) (\frac{39 x}{x - 5})}{7 x + 4 - 7 \cdot x - 5 \cdot - 7}$

Step 5.2.7.3

Multiply $- 5$ by $- 7$ .

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{(x - 5) (\frac{39 x}{x - 5})}{7 x + 4 - 7 \cdot x + 35}$

Step 5.2.7.4

Subtract $7 x$ from $7 x$ .

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{(x - 5) (\frac{39 x}{x - 5})}{0 + 4 + 35}$

Step 5.2.7.5

Add $0$ and $4$ .

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{(x - 5) (\frac{39 x}{x - 5})}{4 + 35}$

Step 5.2.7.6

Add $4$ and $35$ .

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{(x - 5) (\frac{39 x}{x - 5})}{39}$

Step 5.2.8

Reduce the expression by cancelling the common factors.

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Step 5.2.8.1

Factor $\frac{39 x}{x - 5}$ out of $\frac{(x - 5) \frac{39 x}{x - 5}}{39}$ .

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{39 x}{x - 5} \cdot \frac{x - 5}{39}$

Step 5.2.8.2

Cancel the common factor of $39$ .

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Step 5.2.8.2.1

Factor $39$ out of $39 x$ .

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{39 (x)}{x - 5} \cdot \frac{x - 5}{39}$

Step 5.2.8.2.2

Cancel the common factor.

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{39 x}{x - 5} \cdot \frac{x - 5}{39}$

Step 5.2.8.2.3

Rewrite the expression.

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{x}{x - 5} \cdot (x - 5)$

Step 5.2.8.3

Cancel the common factor of $x - 5$ .

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Step 5.2.8.3.1

Cancel the common factor.

$f^{- 1} (\frac{7 x + 4}{x - 5}) = \frac{x}{x - 5} \cdot (x - 5)$

Step 5.2.8.3.2

Rewrite the expression.

$f^{- 1} (\frac{7 x + 4}{x - 5}) = x$

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 $f (\frac{4 + 5 x}{x - 7})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{4 + 5 x}{x - 7}) = \frac{7 (\frac{4 + 5 x}{x - 7}) + 4}{(\frac{4 + 5 x}{x - 7}) - 5}$

Step 5.3.3

Multiply the numerator and denominator of the fraction by $x - 7$ .

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Step 5.3.3.1

Multiply $\frac{7 \frac{4 + 5 x}{x - 7} + 4}{\frac{4 + 5 x}{x - 7} - 5}$ by $\frac{x - 7}{x - 7}$ .

$f (\frac{4 + 5 x}{x - 7}) = \frac{x - 7}{x - 7} \cdot \frac{7 (\frac{4 + 5 x}{x - 7}) + 4}{\frac{4 + 5 x}{x - 7} - 5}$

Step 5.3.3.2

Combine.

$f (\frac{4 + 5 x}{x - 7}) = \frac{(x - 7) (7 (\frac{4 + 5 x}{x - 7}) + 4)}{(x - 7) (\frac{4 + 5 x}{x - 7} - 5)}$

Step 5.3.4

Apply the distributive property.

$f (\frac{4 + 5 x}{x - 7}) = \frac{(x - 7) (7 (\frac{4 + 5 x}{x - 7})) + (x - 7) \cdot 4}{(x - 7) (\frac{4 + 5 x}{x - 7}) + (x - 7) \cdot - 5}$

Step 5.3.5

Cancel the common factor of $x - 7$ .

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Step 5.3.5.1

Cancel the common factor.

$f (\frac{4 + 5 x}{x - 7}) = \frac{(x - 7) (7 (\frac{4 + 5 x}{x - 7})) + (x - 7) \cdot 4}{(x - 7) (\frac{4 + 5 x}{x - 7}) + (x - 7) \cdot - 5}$

Step 5.3.5.2

Rewrite the expression.

$f (\frac{4 + 5 x}{x - 7}) = \frac{(x - 7) (7 (\frac{4 + 5 x}{x - 7})) + (x - 7) \cdot 4}{4 + 5 x + (x - 7) \cdot - 5}$

Step 5.3.6

Simplify the numerator.

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Step 5.3.6.1

Factor $x - 7$ out of $(x - 7) \cdot 7 \frac{4 + 5 x}{x - 7} + (x - 7) \cdot 4$ .

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Step 5.3.6.1.1

Factor $x - 7$ out of $(x - 7) \cdot 7 \frac{4 + 5 x}{x - 7}$ .

$f (\frac{4 + 5 x}{x - 7}) = \frac{(x - 7) (7 (\frac{4 + 5 x}{x - 7})) + (x - 7) \cdot 4}{4 + 5 x + (x - 7) \cdot - 5}$

Step 5.3.6.1.2

Factor $x - 7$ out of $(x - 7) \cdot 4$ .

$f (\frac{4 + 5 x}{x - 7}) = \frac{(x - 7) (7 (\frac{4 + 5 x}{x - 7})) + (x - 7) (4)}{4 + 5 x + (x - 7) \cdot - 5}$

Step 5.3.6.1.3

Factor $x - 7$ out of $(x - 7) (7 \frac{4 + 5 x}{x - 7}) + (x - 7) (4)$ .

$f (\frac{4 + 5 x}{x - 7}) = \frac{(x - 7) (7 (\frac{4 + 5 x}{x - 7}) + 4)}{4 + 5 x + (x - 7) \cdot - 5}$

Step 5.3.6.2

Combine $7$ and $\frac{4 + 5 x}{x - 7}$ .

$f (\frac{4 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{7 (4 + 5 x)}{x - 7} + 4)}{4 + 5 x + (x - 7) \cdot - 5}$

Step 5.3.6.3

To write $4$ as a fraction with a common denominator, multiply by $\frac{x - 7}{x - 7}$ .

$f (\frac{4 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{7 (4 + 5 x)}{x - 7} + \frac{4 (x - 7)}{x - 7})}{4 + 5 x + (x - 7) \cdot - 5}$

Step 5.3.6.4

Combine the numerators over the common denominator.

$f (\frac{4 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{7 (4 + 5 x) + 4 (x - 7)}{x - 7})}{4 + 5 x + (x - 7) \cdot - 5}$

Step 5.3.6.5

Rewrite $\frac{7 (4 + 5 x) + 4 (x - 7)}{x - 7}$ in a factored form.

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Step 5.3.6.5.1

Apply the distributive property.

$f (\frac{4 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{7 \cdot 4 + 7 (5 x) + 4 (x - 7)}{x - 7})}{4 + 5 x + (x - 7) \cdot - 5}$

Step 5.3.6.5.2

Multiply $7$ by $4$ .

$f (\frac{4 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{28 + 7 (5 x) + 4 (x - 7)}{x - 7})}{4 + 5 x + (x - 7) \cdot - 5}$

Step 5.3.6.5.3

Multiply $5$ by $7$ .

$f (\frac{4 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{28 + 35 x + 4 (x - 7)}{x - 7})}{4 + 5 x + (x - 7) \cdot - 5}$

Step 5.3.6.5.4

Apply the distributive property.

$f (\frac{4 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{28 + 35 x + 4 x + 4 \cdot - 7}{x - 7})}{4 + 5 x + (x - 7) \cdot - 5}$

Step 5.3.6.5.5

Multiply $4$ by $- 7$ .

$f (\frac{4 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{28 + 35 x + 4 x - 28}{x - 7})}{4 + 5 x + (x - 7) \cdot - 5}$

Step 5.3.6.5.6

Subtract $28$ from $28$ .

$f (\frac{4 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{35 x + 4 x + 0}{x - 7})}{4 + 5 x + (x - 7) \cdot - 5}$

Step 5.3.6.5.7

Add $35 x + 4 x$ and $0$ .

$f (\frac{4 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{35 x + 4 x}{x - 7})}{4 + 5 x + (x - 7) \cdot - 5}$

Step 5.3.6.5.8

Add $35 x$ and $4 x$ .

$f (\frac{4 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{39 x}{x - 7})}{4 + 5 x + (x - 7) \cdot - 5}$

Step 5.3.7

Simplify the denominator.

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Step 5.3.7.1

Apply the distributive property.

$f (\frac{4 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{39 x}{x - 7})}{4 + 5 x + x \cdot - 5 - 7 \cdot - 5}$

Step 5.3.7.2

Move $- 5$ to the left of $x$ .

$f (\frac{4 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{39 x}{x - 7})}{4 + 5 x - 5 \cdot x - 7 \cdot - 5}$

Step 5.3.7.3

Multiply $- 7$ by $- 5$ .

$f (\frac{4 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{39 x}{x - 7})}{4 + 5 x - 5 \cdot x + 35}$

Step 5.3.7.4

Add $4$ and $35$ .

$f (\frac{4 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{39 x}{x - 7})}{5 x - 5 x + 39}$

Step 5.3.7.5

Subtract $5 x$ from $5 x$ .

$f (\frac{4 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{39 x}{x - 7})}{0 + 39}$

Step 5.3.7.6

Add $0$ and $39$ .

$f (\frac{4 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{39 x}{x - 7})}{39}$

Step 5.3.8

Reduce the expression by cancelling the common factors.

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Step 5.3.8.1

Factor $\frac{39 x}{x - 7}$ out of $\frac{(x - 7) \frac{39 x}{x - 7}}{39}$ .

$f (\frac{4 + 5 x}{x - 7}) = \frac{39 x}{x - 7} \cdot \frac{x - 7}{39}$

Step 5.3.8.2

Cancel the common factor of $39$ .

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Step 5.3.8.2.1

Factor $39$ out of $39 x$ .

$f (\frac{4 + 5 x}{x - 7}) = \frac{39 (x)}{x - 7} \cdot \frac{x - 7}{39}$

Step 5.3.8.2.2

Cancel the common factor.

$f (\frac{4 + 5 x}{x - 7}) = \frac{39 x}{x - 7} \cdot \frac{x - 7}{39}$

Step 5.3.8.2.3

Rewrite the expression.

$f (\frac{4 + 5 x}{x - 7}) = \frac{x}{x - 7} \cdot (x - 7)$

Step 5.3.8.3

Cancel the common factor of $x - 7$ .

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Step 5.3.8.3.1

Cancel the common factor.

$f (\frac{4 + 5 x}{x - 7}) = \frac{x}{x - 7} \cdot (x - 7)$

Step 5.3.8.3.2

Rewrite the expression.

$f (\frac{4 + 5 x}{x - 7}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{4 + 5 x}{x - 7}$ is the inverse of $f (x) = \frac{7 x + 4}{x - 5}$ .

$f^{- 1} (x) = \frac{4 + 5 x}{x - 7}$