Finite Math Examples

$y = - \frac{4}{5} x + 2$

Step 1

Interchange the variables.

$x = - \frac{4}{5} y + 2$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $- \frac{4}{5} y + 2 = x$ .

$- \frac{4}{5} y + 2 = x$

Step 2.2

Simplify each term.

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Step 2.2.1

Combine $y$ and $\frac{4}{5}$ .

$- \frac{y \cdot 4}{5} + 2 = x$

Step 2.2.2

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

$- \frac{4 y}{5} + 2 = x$

Step 2.3

Subtract $2$ from both sides of the equation.

$- \frac{4 y}{5} = x - 2$

Step 2.4

Multiply both sides of the equation by $- \frac{5}{4}$ .

$- \frac{5}{4} (- \frac{4 y}{5}) = - \frac{5}{4} (x - 2)$

Step 2.5

Simplify both sides of the equation.

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Step 2.5.1

Simplify the left side.

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Step 2.5.1.1

Simplify $- \frac{5}{4} (- \frac{4 y}{5})$ .

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Step 2.5.1.1.1

Cancel the common factor of $5$ .

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Step 2.5.1.1.1.1

Move the leading negative in $- \frac{5}{4}$ into the numerator.

$\frac{- 5}{4} (- \frac{4 y}{5}) = - \frac{5}{4} (x - 2)$

Step 2.5.1.1.1.2

Move the leading negative in $- \frac{4 y}{5}$ into the numerator.

$\frac{- 5}{4} \cdot \frac{- 4 y}{5} = - \frac{5}{4} (x - 2)$

Step 2.5.1.1.1.3

Factor $5$ out of $- 5$ .

$\frac{5 (- 1)}{4} \cdot \frac{- 4 y}{5} = - \frac{5}{4} (x - 2)$

Step 2.5.1.1.1.4

Cancel the common factor.

$\frac{5 \cdot - 1}{4} \cdot \frac{- 4 y}{5} = - \frac{5}{4} (x - 2)$

Step 2.5.1.1.1.5

Rewrite the expression.

$\frac{- 1}{4} (- 4 y) = - \frac{5}{4} (x - 2)$

Step 2.5.1.1.2

Cancel the common factor of $4$ .

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Step 2.5.1.1.2.1

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

$\frac{- 1}{4} (4 (- y)) = - \frac{5}{4} (x - 2)$

Step 2.5.1.1.2.2

Cancel the common factor.

$\frac{- 1}{4} (4 (- y)) = - \frac{5}{4} (x - 2)$

Step 2.5.1.1.2.3

Rewrite the expression.

$- - y = - \frac{5}{4} (x - 2)$

Step 2.5.1.1.3

Multiply.

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Step 2.5.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{5}{4} (x - 2)$

Step 2.5.1.1.3.2

Multiply $y$ by $1$ .

$y = - \frac{5}{4} (x - 2)$

Step 2.5.2

Simplify the right side.

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Step 2.5.2.1

Simplify $- \frac{5}{4} (x - 2)$ .

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Step 2.5.2.1.1

Simplify terms.

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Step 2.5.2.1.1.1

Apply the distributive property.

$y = - \frac{5}{4} x - \frac{5}{4} \cdot - 2$

Step 2.5.2.1.1.2

Combine $x$ and $\frac{5}{4}$ .

$y = - \frac{x \cdot 5}{4} - \frac{5}{4} \cdot - 2$

Step 2.5.2.1.1.3

Cancel the common factor of $2$ .

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Step 2.5.2.1.1.3.1

Move the leading negative in $- \frac{5}{4}$ into the numerator.

$y = - \frac{x \cdot 5}{4} + \frac{- 5}{4} \cdot - 2$

Step 2.5.2.1.1.3.2

Factor $2$ out of $4$ .

$y = - \frac{x \cdot 5}{4} + \frac{- 5}{2 (2)} \cdot - 2$

Step 2.5.2.1.1.3.3

Factor $2$ out of $- 2$ .

$y = - \frac{x \cdot 5}{4} + \frac{- 5}{2 \cdot 2} \cdot (2 \cdot - 1)$

Step 2.5.2.1.1.3.4

Cancel the common factor.

$y = - \frac{x \cdot 5}{4} + \frac{- 5}{2 \cdot 2} \cdot (2 \cdot - 1)$

Step 2.5.2.1.1.3.5

Rewrite the expression.

$y = - \frac{x \cdot 5}{4} + \frac{- 5}{2} \cdot - 1$

Step 2.5.2.1.1.4

Combine $\frac{- 5}{2}$ and $- 1$ .

$y = - \frac{x \cdot 5}{4} + \frac{- 5 \cdot - 1}{2}$

Step 2.5.2.1.1.5

Multiply $- 5$ by $- 1$ .

$y = - \frac{x \cdot 5}{4} + \frac{5}{2}$

Step 2.5.2.1.2

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

$y = - \frac{5 x}{4} + \frac{5}{2}$

Step 3

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

$f^{- 1} (x) = - \frac{5 x}{4} + \frac{5}{2}$

Step 4

Verify if $f^{- 1} (x) = - \frac{5 x}{4} + \frac{5}{2}$ is the inverse of $f (x) = - \frac{4}{5} x + 2$ .

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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 $f^{- 1} (- \frac{4}{5} x + 2)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \frac{4}{5} x + 2) = - \frac{5 (- \frac{4}{5} x + 2)}{4} + \frac{5}{2}$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Simplify the numerator.

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Step 4.2.3.1.1

Combine $x$ and $\frac{4}{5}$ .

$f^{- 1} (- \frac{4}{5} x + 2) = - \frac{5 (- \frac{x \cdot 4}{5} + 2)}{4} + \frac{5}{2}$

Step 4.2.3.1.2

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

$f^{- 1} (- \frac{4}{5} x + 2) = - \frac{5 (- \frac{4 \cdot x}{5} + 2)}{4} + \frac{5}{2}$

Step 4.2.3.1.3

Factor $2$ out of $- \frac{4 x}{5} + 2$ .

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Step 4.2.3.1.3.1

Factor $2$ out of $- \frac{4 x}{5}$ .

$f^{- 1} (- \frac{4}{5} x + 2) = - \frac{5 (2 (- \frac{2 x}{5}) + 2)}{4} + \frac{5}{2}$

Step 4.2.3.1.3.2

Factor $2$ out of $2$ .

$f^{- 1} (- \frac{4}{5} x + 2) = - \frac{5 (2 (- \frac{2 x}{5}) + 2 (1))}{4} + \frac{5}{2}$

Step 4.2.3.1.3.3

Factor $2$ out of $2 (- \frac{2 x}{5}) + 2 (1)$ .

$f^{- 1} (- \frac{4}{5} x + 2) = - \frac{5 (2 (- \frac{2 x}{5} + 1))}{4} + \frac{5}{2}$

$f^{- 1} (- \frac{4}{5} x + 2) = - \frac{5 \cdot (2 (- \frac{2 x}{5} + 1))}{4} + \frac{5}{2}$

Step 4.2.3.1.4

Multiply $5$ by $2$ .

$f^{- 1} (- \frac{4}{5} x + 2) = - \frac{10 (- \frac{2 x}{5} + 1)}{4} + \frac{5}{2}$

Step 4.2.3.1.5

Write $1$ as a fraction with a common denominator.

$f^{- 1} (- \frac{4}{5} x + 2) = - \frac{10 (- \frac{2 x}{5} + \frac{5}{5})}{4} + \frac{5}{2}$

Step 4.2.3.1.6

Combine the numerators over the common denominator.

$f^{- 1} (- \frac{4}{5} x + 2) = - \frac{10 (\frac{- 2 x + 5}{5})}{4} + \frac{5}{2}$

Step 4.2.3.2

Combine $10$ and $\frac{- 2 x + 5}{5}$ .

$f^{- 1} (- \frac{4}{5} x + 2) = - \frac{\frac{10 (- 2 x + 5)}{5}}{4} + \frac{5}{2}$

Step 4.2.3.3

Reduce the expression by cancelling the common factors.

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Step 4.2.3.3.1

Reduce the expression $\frac{10 (- 2 x + 5)}{5}$ by cancelling the common factors.

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Step 4.2.3.3.1.1

Factor $5$ out of $10 (- 2 x + 5)$ .

$f^{- 1} (- \frac{4}{5} x + 2) = - \frac{\frac{5 (2 (- 2 x + 5))}{5}}{4} + \frac{5}{2}$

Step 4.2.3.3.1.2

Factor $5$ out of $5$ .

$f^{- 1} (- \frac{4}{5} x + 2) = - \frac{\frac{5 (2 (- 2 x + 5))}{5 (1)}}{4} + \frac{5}{2}$

Step 4.2.3.3.1.3

Cancel the common factor.

$f^{- 1} (- \frac{4}{5} x + 2) = - \frac{\frac{5 (2 (- 2 x + 5))}{5 \cdot 1}}{4} + \frac{5}{2}$

Step 4.2.3.3.1.4

Rewrite the expression.

$f^{- 1} (- \frac{4}{5} x + 2) = - \frac{\frac{2 (- 2 x + 5)}{1}}{4} + \frac{5}{2}$

Step 4.2.3.3.2

Divide $2 (- 2 x + 5)$ by $1$ .

$f^{- 1} (- \frac{4}{5} x + 2) = - \frac{2 (- 2 x + 5)}{4} + \frac{5}{2}$

Step 4.2.3.4

Cancel the common factors.

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Step 4.2.3.4.1

Factor $2$ out of $4$ .

$f^{- 1} (- \frac{4}{5} x + 2) = - \frac{2 (- 2 x + 5)}{2 \cdot 2} + \frac{5}{2}$

Step 4.2.3.4.2

Cancel the common factor.

$f^{- 1} (- \frac{4}{5} x + 2) = - \frac{2 (- 2 x + 5)}{2 \cdot 2} + \frac{5}{2}$

Step 4.2.3.4.3

Rewrite the expression.

$f^{- 1} (- \frac{4}{5} x + 2) = - \frac{- 2 x + 5}{2} + \frac{5}{2}$

Step 4.2.4

Combine the numerators over the common denominator.

$f^{- 1} (- \frac{4}{5} x + 2) = \frac{- (- 2 x + 5) + 5}{2}$

Step 4.2.5

Simplify each term.

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Step 4.2.5.1

Apply the distributive property.

$f^{- 1} (- \frac{4}{5} x + 2) = \frac{- (- 2 x) - 1 \cdot 5 + 5}{2}$

Step 4.2.5.2

Multiply $- 2$ by $- 1$ .

$f^{- 1} (- \frac{4}{5} x + 2) = \frac{2 x - 1 \cdot 5 + 5}{2}$

Step 4.2.5.3

Multiply $- 1$ by $5$ .

$f^{- 1} (- \frac{4}{5} x + 2) = \frac{2 x - 5 + 5}{2}$

Step 4.2.6

Simplify terms.

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Step 4.2.6.1

Combine the opposite terms in $2 x - 5 + 5$ .

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Step 4.2.6.1.1

Add $- 5$ and $5$ .

$f^{- 1} (- \frac{4}{5} x + 2) = \frac{2 x + 0}{2}$

Step 4.2.6.1.2

Add $2 x$ and $0$ .

$f^{- 1} (- \frac{4}{5} x + 2) = \frac{2 x}{2}$

Step 4.2.6.2

Cancel the common factor of $2$ .

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Step 4.2.6.2.1

Cancel the common factor.

$f^{- 1} (- \frac{4}{5} x + 2) = \frac{2 x}{2}$

Step 4.2.6.2.2

Divide $x$ by $1$ .

$f^{- 1} (- \frac{4}{5} x + 2) = x$

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 $f (- \frac{5 x}{4} + \frac{5}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{5 x}{4} + \frac{5}{2}) = - \frac{4}{5} \cdot (- \frac{5 x}{4} + \frac{5}{2}) + 2$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Apply the distributive property.

$f (- \frac{5 x}{4} + \frac{5}{2}) = - \frac{4}{5} \cdot (- \frac{5 x}{4}) - \frac{4}{5} \cdot \frac{5}{2} + 2$

Step 4.3.3.2

Cancel the common factor of $4$ .

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Step 4.3.3.2.1

Move the leading negative in $- \frac{4}{5}$ into the numerator.

$f (- \frac{5 x}{4} + \frac{5}{2}) = \frac{- 4}{5} \cdot (- \frac{5 x}{4}) - \frac{4}{5} \cdot \frac{5}{2} + 2$

Step 4.3.3.2.2

Move the leading negative in $- \frac{5 x}{4}$ into the numerator.

$f (- \frac{5 x}{4} + \frac{5}{2}) = \frac{- 4}{5} \cdot \frac{- 5 x}{4} - \frac{4}{5} \cdot \frac{5}{2} + 2$

Step 4.3.3.2.3

Factor $4$ out of $- 4$ .

$f (- \frac{5 x}{4} + \frac{5}{2}) = \frac{4 (- 1)}{5} \cdot \frac{- 5 x}{4} - \frac{4}{5} \cdot \frac{5}{2} + 2$

Step 4.3.3.2.4

Cancel the common factor.

$f (- \frac{5 x}{4} + \frac{5}{2}) = \frac{4 \cdot - 1}{5} \cdot \frac{- 5 x}{4} - \frac{4}{5} \cdot \frac{5}{2} + 2$

Step 4.3.3.2.5

Rewrite the expression.

$f (- \frac{5 x}{4} + \frac{5}{2}) = \frac{- 1}{5} \cdot (- 5 x) - \frac{4}{5} \cdot \frac{5}{2} + 2$

Step 4.3.3.3

Cancel the common factor of $5$ .

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Step 4.3.3.3.1

Factor $5$ out of $- 5 x$ .

$f (- \frac{5 x}{4} + \frac{5}{2}) = \frac{- 1}{5} \cdot (5 (- x)) - \frac{4}{5} \cdot \frac{5}{2} + 2$

Step 4.3.3.3.2

Cancel the common factor.

$f (- \frac{5 x}{4} + \frac{5}{2}) = \frac{- 1}{5} \cdot (5 (- x)) - \frac{4}{5} \cdot \frac{5}{2} + 2$

Step 4.3.3.3.3

Rewrite the expression.

$f (- \frac{5 x}{4} + \frac{5}{2}) = - 1 (- x) - \frac{4}{5} \cdot \frac{5}{2} + 2$

Step 4.3.3.4

Multiply $- 1$ by $- 1$ .

$f (- \frac{5 x}{4} + \frac{5}{2}) = 1 x - \frac{4}{5} \cdot \frac{5}{2} + 2$

Step 4.3.3.5

Multiply $x$ by $1$ .

$f (- \frac{5 x}{4} + \frac{5}{2}) = x - \frac{4}{5} \cdot \frac{5}{2} + 2$

Step 4.3.3.6

Cancel the common factor of $2$ .

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Step 4.3.3.6.1

Move the leading negative in $- \frac{4}{5}$ into the numerator.

$f (- \frac{5 x}{4} + \frac{5}{2}) = x + \frac{- 4}{5} \cdot \frac{5}{2} + 2$

Step 4.3.3.6.2

Factor $2$ out of $- 4$ .

$f (- \frac{5 x}{4} + \frac{5}{2}) = x + \frac{2 (- 2)}{5} \cdot \frac{5}{2} + 2$

Step 4.3.3.6.3

Cancel the common factor.

$f (- \frac{5 x}{4} + \frac{5}{2}) = x + \frac{2 \cdot - 2}{5} \cdot \frac{5}{2} + 2$

Step 4.3.3.6.4

Rewrite the expression.

$f (- \frac{5 x}{4} + \frac{5}{2}) = x + \frac{- 2}{5} \cdot 5 + 2$

Step 4.3.3.7

Cancel the common factor of $5$ .

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Step 4.3.3.7.1

Cancel the common factor.

$f (- \frac{5 x}{4} + \frac{5}{2}) = x + \frac{- 2}{5} \cdot 5 + 2$

Step 4.3.3.7.2

Rewrite the expression.

$f (- \frac{5 x}{4} + \frac{5}{2}) = x - 2 + 2$

Step 4.3.4

Combine the opposite terms in $x - 2 + 2$ .

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Step 4.3.4.1

Add $- 2$ and $2$ .

$f (- \frac{5 x}{4} + \frac{5}{2}) = x + 0$

Step 4.3.4.2

Add $x$ and $0$ .

$f (- \frac{5 x}{4} + \frac{5}{2}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{5 x}{4} + \frac{5}{2}$ is the inverse of $f (x) = - \frac{4}{5} x + 2$ .

$f^{- 1} (x) = - \frac{5 x}{4} + \frac{5}{2}$