Finite Math Examples

$f (x) = \frac{- 4}{7} x - \frac{16}{7}$

Step 1

Write $f (x) = \frac{- 4}{7} x - \frac{16}{7}$ as an equation.

$y = \frac{- 4}{7} x - \frac{16}{7}$

Step 2

Interchange the variables.

$x = \frac{- 4}{7} y - \frac{16}{7}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{- 4}{7} y - \frac{16}{7} = x$ .

$\frac{- 4}{7} y - \frac{16}{7} = x$

Step 3.2

Simplify each term.

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Step 3.2.1

Move the negative in front of the fraction.

$- \frac{4}{7} y - \frac{16}{7} = x$

Step 3.2.2

Combine $y$ and $\frac{4}{7}$ .

$- \frac{y \cdot 4}{7} - \frac{16}{7} = x$

Step 3.2.3

Move $4$ to the left of $y$ .

$- \frac{4 y}{7} - \frac{16}{7} = x$

Step 3.3

Add $\frac{16}{7}$ to both sides of the equation.

$- \frac{4 y}{7} = x + \frac{16}{7}$

Step 3.4

Multiply both sides of the equation by $- \frac{7}{4}$ .

$- \frac{7}{4} (- \frac{4 y}{7}) = - \frac{7}{4} (x + \frac{16}{7})$

Step 3.5

Simplify both sides of the equation.

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Step 3.5.1

Simplify the left side.

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Step 3.5.1.1

Simplify $- \frac{7}{4} (- \frac{4 y}{7})$ .

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Step 3.5.1.1.1

Cancel the common factor of $7$ .

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Step 3.5.1.1.1.1

Move the leading negative in $- \frac{7}{4}$ into the numerator.

$\frac{- 7}{4} (- \frac{4 y}{7}) = - \frac{7}{4} (x + \frac{16}{7})$

Step 3.5.1.1.1.2

Move the leading negative in $- \frac{4 y}{7}$ into the numerator.

$\frac{- 7}{4} \cdot \frac{- 4 y}{7} = - \frac{7}{4} (x + \frac{16}{7})$

Step 3.5.1.1.1.3

Factor $7$ out of $- 7$ .

$\frac{7 (- 1)}{4} \cdot \frac{- 4 y}{7} = - \frac{7}{4} (x + \frac{16}{7})$

Step 3.5.1.1.1.4

Cancel the common factor.

$\frac{7 \cdot - 1}{4} \cdot \frac{- 4 y}{7} = - \frac{7}{4} (x + \frac{16}{7})$

Step 3.5.1.1.1.5

Rewrite the expression.

$\frac{- 1}{4} (- 4 y) = - \frac{7}{4} (x + \frac{16}{7})$

Step 3.5.1.1.2

Cancel the common factor of $4$ .

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Step 3.5.1.1.2.1

Factor $4$ out of $- 4 y$ .

$\frac{- 1}{4} (4 (- y)) = - \frac{7}{4} (x + \frac{16}{7})$

Step 3.5.1.1.2.2

Cancel the common factor.

$\frac{- 1}{4} (4 (- y)) = - \frac{7}{4} (x + \frac{16}{7})$

Step 3.5.1.1.2.3

Rewrite the expression.

$- - y = - \frac{7}{4} (x + \frac{16}{7})$

Step 3.5.1.1.3

Multiply.

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Step 3.5.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{7}{4} (x + \frac{16}{7})$

Step 3.5.1.1.3.2

Multiply $y$ by $1$ .

$y = - \frac{7}{4} (x + \frac{16}{7})$

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Simplify $- \frac{7}{4} (x + \frac{16}{7})$ .

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Step 3.5.2.1.1

Simplify terms.

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Step 3.5.2.1.1.1

Apply the distributive property.

$y = - \frac{7}{4} x - \frac{7}{4} \cdot \frac{16}{7}$

Step 3.5.2.1.1.2

Combine $x$ and $\frac{7}{4}$ .

$y = - \frac{x \cdot 7}{4} - \frac{7}{4} \cdot \frac{16}{7}$

Step 3.5.2.1.1.3

Cancel the common factor of $7$ .

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Step 3.5.2.1.1.3.1

Move the leading negative in $- \frac{7}{4}$ into the numerator.

$y = - \frac{x \cdot 7}{4} + \frac{- 7}{4} \cdot \frac{16}{7}$

Step 3.5.2.1.1.3.2

Factor $7$ out of $- 7$ .

$y = - \frac{x \cdot 7}{4} + \frac{7 (- 1)}{4} \cdot \frac{16}{7}$

Step 3.5.2.1.1.3.3

Cancel the common factor.

$y = - \frac{x \cdot 7}{4} + \frac{7 \cdot - 1}{4} \cdot \frac{16}{7}$

Step 3.5.2.1.1.3.4

Rewrite the expression.

$y = - \frac{x \cdot 7}{4} + \frac{- 1}{4} \cdot 16$

Step 3.5.2.1.1.4

Cancel the common factor of $4$ .

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Step 3.5.2.1.1.4.1

Factor $4$ out of $16$ .

$y = - \frac{x \cdot 7}{4} + \frac{- 1}{4} \cdot (4 (4))$

Step 3.5.2.1.1.4.2

Cancel the common factor.

$y = - \frac{x \cdot 7}{4} + \frac{- 1}{4} \cdot (4 \cdot 4)$

Step 3.5.2.1.1.4.3

Rewrite the expression.

$y = - \frac{x \cdot 7}{4} - 1 \cdot 4$

Step 3.5.2.1.1.5

Multiply $- 1$ by $4$ .

$y = - \frac{x \cdot 7}{4} - 4$

Step 3.5.2.1.2

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

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

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = - \frac{7 x}{4} - 4$ is the inverse of $f (x) = \frac{- 4}{7} x - \frac{16}{7}$ .

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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 $f^{- 1} (\frac{- 4}{7} x - \frac{16}{7})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = - \frac{7 (\frac{- 4}{7} x - \frac{16}{7})}{4} - 4$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Simplify the numerator.

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Step 5.2.3.1.1

Move the negative in front of the fraction.

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = - \frac{7 (- \frac{4}{7} x - \frac{16}{7})}{4} - 4$

Step 5.2.3.1.2

Combine $x$ and $\frac{4}{7}$ .

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = - \frac{7 (- \frac{x \cdot 4}{7} - \frac{16}{7})}{4} - 4$

Step 5.2.3.1.3

Move $4$ to the left of $x$ .

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = - \frac{7 (- \frac{4 \cdot x}{7} - \frac{16}{7})}{4} - 4$

Step 5.2.3.1.4

Factor $- 1$ out of $- \frac{4 x}{7} - \frac{16}{7}$ .

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Step 5.2.3.1.4.1

Factor $- 1$ out of $- \frac{4 x}{7}$ .

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = - \frac{7 (- (\frac{4 x}{7}) - \frac{16}{7})}{4} - 4$

Step 5.2.3.1.4.2

Factor $- 1$ out of $- \frac{16}{7}$ .

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = - \frac{7 (- (\frac{4 x}{7}) - (\frac{16}{7}))}{4} - 4$

Step 5.2.3.1.4.3

Factor $- 1$ out of $- (\frac{4 x}{7}) - (\frac{16}{7})$ .

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = - \frac{7 (- (\frac{4 x}{7} + \frac{16}{7}))}{4} - 4$

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = - \frac{7 \cdot (- 1 (\frac{4 x}{7} + \frac{16}{7}))}{4} - 4$

Step 5.2.3.1.5

Combine exponents.

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Step 5.2.3.1.5.1

Factor out negative.

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = - \frac{- (7 (\frac{4 x}{7} + \frac{16}{7}))}{4} - 4$

Step 5.2.3.1.5.2

Multiply $7$ by $- 1$ .

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = - \frac{- 7 (\frac{4 x}{7} + \frac{16}{7})}{4} - 4$

Step 5.2.3.1.6

Combine the numerators over the common denominator.

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = - \frac{- 7 \frac{4 x + 16}{7}}{4} - 4$

Step 5.2.3.1.7

Factor $4$ out of $4 x + 16$ .

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Step 5.2.3.1.7.1

Factor $4$ out of $4 x$ .

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = - \frac{- 7 \frac{4 (x) + 16}{7}}{4} - 4$

Step 5.2.3.1.7.2

Factor $4$ out of $16$ .

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = - \frac{- 7 \frac{4 x + 4 \cdot 4}{7}}{4} - 4$

Step 5.2.3.1.7.3

Factor $4$ out of $4 x + 4 \cdot 4$ .

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = - \frac{- 7 \frac{4 (x + 4)}{7}}{4} - 4$

Step 5.2.3.2

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

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = - \frac{\frac{- 7 (4 (x + 4))}{7}}{4} - 4$

Step 5.2.3.3

Multiply $- 7$ by $4$ .

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = - \frac{\frac{- 28 (x + 4)}{7}}{4} - 4$

Step 5.2.3.4

Reduce the expression by cancelling the common factors.

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Step 5.2.3.4.1

Reduce the expression $\frac{- 28 (x + 4)}{7}$ by cancelling the common factors.

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Step 5.2.3.4.1.1

Factor $7$ out of $- 28 (x + 4)$ .

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = - \frac{\frac{7 (- 4 (x + 4))}{7}}{4} - 4$

Step 5.2.3.4.1.2

Factor $7$ out of $7$ .

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = - \frac{\frac{7 (- 4 (x + 4))}{7 (1)}}{4} - 4$

Step 5.2.3.4.1.3

Cancel the common factor.

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = - \frac{\frac{7 (- 4 (x + 4))}{7 \cdot 1}}{4} - 4$

Step 5.2.3.4.1.4

Rewrite the expression.

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = - \frac{\frac{- 4 (x + 4)}{1}}{4} - 4$

Step 5.2.3.4.2

Divide $- 4 (x + 4)$ by $1$ .

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = - \frac{- 4 (x + 4)}{4} - 4$

Step 5.2.3.5

Cancel the common factor of $- 4$ and $4$ .

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Step 5.2.3.5.1

Factor $4$ out of $- 4 (x + 4)$ .

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = - \frac{4 (- (x + 4))}{4} - 4$

Step 5.2.3.5.2

Cancel the common factors.

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Step 5.2.3.5.2.1

Factor $4$ out of $4$ .

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = - \frac{4 (- (x + 4))}{4 (1)} - 4$

Step 5.2.3.5.2.2

Cancel the common factor.

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = - \frac{4 (- (x + 4))}{4 \cdot 1} - 4$

Step 5.2.3.5.2.3

Rewrite the expression.

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = - \frac{- (x + 4)}{1} - 4$

Step 5.2.3.5.2.4

Divide $- (x + 4)$ by $1$ .

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = (x + 4) - 4$

Step 5.2.3.6

Apply the distributive property.

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = - (- x - 1 \cdot 4) - 4$

Step 5.2.3.7

Multiply $- 1$ by $4$ .

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = - (- x - 4) - 4$

Step 5.2.3.8

Apply the distributive property.

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = x + 4 - 4$

Step 5.2.3.9

Multiply $- - x$ .

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Step 5.2.3.9.1

Multiply $- 1$ by $- 1$ .

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = 1 x + 4 - 4$

Step 5.2.3.9.2

Multiply $x$ by $1$ .

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = x + 4 - 4$

Step 5.2.3.10

Multiply $- 1$ by $- 4$ .

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = x + 4 - 4$

Step 5.2.4

Combine the opposite terms in $x + 4 - 4$ .

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Step 5.2.4.1

Subtract $4$ from $4$ .

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} (\frac{- 4}{7} x - \frac{16}{7}) = x$

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

$f (- \frac{7 x}{4} - 4) = \frac{- 4}{7} \cdot (- \frac{7 x}{4} - 4) - \frac{16}{7}$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Move the negative in front of the fraction.

$f (- \frac{7 x}{4} - 4) = - \frac{4}{7} \cdot (- \frac{7 x}{4} - 4) - \frac{16}{7}$

Step 5.3.3.2

Apply the distributive property.

$f (- \frac{7 x}{4} - 4) = - \frac{4}{7} \cdot (- \frac{7 x}{4}) - \frac{4}{7} \cdot - 4 - \frac{16}{7}$

Step 5.3.3.3

Cancel the common factor of $4$ .

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Step 5.3.3.3.1

Move the leading negative in $- \frac{4}{7}$ into the numerator.

$f (- \frac{7 x}{4} - 4) = \frac{- 4}{7} \cdot (- \frac{7 x}{4}) - \frac{4}{7} \cdot - 4 - \frac{16}{7}$

Step 5.3.3.3.2

Move the leading negative in $- \frac{7 x}{4}$ into the numerator.

$f (- \frac{7 x}{4} - 4) = \frac{- 4}{7} \cdot \frac{- 7 x}{4} - \frac{4}{7} \cdot - 4 - \frac{16}{7}$

Step 5.3.3.3.3

Factor $4$ out of $- 4$ .

$f (- \frac{7 x}{4} - 4) = \frac{4 (- 1)}{7} \cdot \frac{- 7 x}{4} - \frac{4}{7} \cdot - 4 - \frac{16}{7}$

Step 5.3.3.3.4

Cancel the common factor.

$f (- \frac{7 x}{4} - 4) = \frac{4 \cdot - 1}{7} \cdot \frac{- 7 x}{4} - \frac{4}{7} \cdot - 4 - \frac{16}{7}$

Step 5.3.3.3.5

Rewrite the expression.

$f (- \frac{7 x}{4} - 4) = \frac{- 1}{7} \cdot (- 7 x) - \frac{4}{7} \cdot - 4 - \frac{16}{7}$

Step 5.3.3.4

Cancel the common factor of $7$ .

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Step 5.3.3.4.1

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

$f (- \frac{7 x}{4} - 4) = \frac{- 1}{7} \cdot (7 (- x)) - \frac{4}{7} \cdot - 4 - \frac{16}{7}$

Step 5.3.3.4.2

Cancel the common factor.

$f (- \frac{7 x}{4} - 4) = \frac{- 1}{7} \cdot (7 (- x)) - \frac{4}{7} \cdot - 4 - \frac{16}{7}$

Step 5.3.3.4.3

Rewrite the expression.

$f (- \frac{7 x}{4} - 4) = - 1 (- x) - \frac{4}{7} \cdot - 4 - \frac{16}{7}$

Step 5.3.3.5

Multiply $- 1$ by $- 1$ .

$f (- \frac{7 x}{4} - 4) = 1 x - \frac{4}{7} \cdot - 4 - \frac{16}{7}$

Step 5.3.3.6

Multiply $x$ by $1$ .

$f (- \frac{7 x}{4} - 4) = x - \frac{4}{7} \cdot - 4 - \frac{16}{7}$

Step 5.3.3.7

Multiply $- \frac{4}{7} \cdot - 4$ .

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Step 5.3.3.7.1

Multiply $- 4$ by $- 1$ .

$f (- \frac{7 x}{4} - 4) = x + 4 (\frac{4}{7}) - \frac{16}{7}$

Step 5.3.3.7.2

Combine $4$ and $\frac{4}{7}$ .

$f (- \frac{7 x}{4} - 4) = x + \frac{4 \cdot 4}{7} - \frac{16}{7}$

Step 5.3.3.7.3

Multiply $4$ by $4$ .

$f (- \frac{7 x}{4} - 4) = x + \frac{16}{7} - \frac{16}{7}$

Step 5.3.4

Combine the opposite terms in $x + \frac{16}{7} - \frac{16}{7}$ .

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Step 5.3.4.1

Combine the numerators over the common denominator.

$f (- \frac{7 x}{4} - 4) = x + \frac{16 - 16}{7}$

Step 5.3.4.2

Subtract $16$ from $16$ .

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

Step 5.3.4.3

Divide $0$ by $7$ .

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

Step 5.3.4.4

Add $x$ and $0$ .

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

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{7 x}{4} - 4$ is the inverse of $f (x) = \frac{- 4}{7} x - \frac{16}{7}$ .

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