Finite Math Examples

$f (x) = 8 x + 7 y$

Step 1

Write $f (x) = 8 x + 7 y$ as an equation.

$x = 8 x + 7 y$

Step 2

Rewrite the equation as $8 x + 7 y = x$ .

$8 x + 7 y = x$

Step 3

Move all terms not containing $y$ to the right side of the equation.

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Step 3.1

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

$7 y = x - 8 x$

Step 3.2

Subtract $8 x$ from $x$ .

$7 y = - 7 x$

Step 4

Divide each term in $7 y = - 7 x$ by $7$ and simplify.

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Step 4.1

Divide each term in $7 y = - 7 x$ by $7$ .

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $7$ .

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Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Divide $y$ by $1$ .

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

Step 4.3

Simplify the right side.

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Step 4.3.1

Cancel the common factor of $- 7$ and $7$ .

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Step 4.3.1.1

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

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

Step 4.3.1.2

Cancel the common factors.

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Step 4.3.1.2.1

Factor $7$ out of $7$ .

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

Step 4.3.1.2.2

Cancel the common factor.

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

Step 4.3.1.2.3

Rewrite the expression.

$y = \frac{- x}{1}$

Step 4.3.1.2.4

Divide $- x$ by $1$ .

$y = - x$

Step 5

Interchange the variables.

$x = - y$

Step 6

Solve for $y$ .

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Step 6.1

Rewrite the equation as $- y = x$ .

$- y = x$

Step 6.2

Divide each term in $- y = x$ by $- 1$ and simplify.

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Step 6.2.1

Divide each term in $- y = x$ by $- 1$ .

$\frac{- y}{- 1} = \frac{x}{- 1}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{x}{- 1}$

Step 6.2.2.2

Divide $y$ by $1$ .

$y = \frac{x}{- 1}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Move the negative one from the denominator of $\frac{x}{- 1}$ .

$y = - 1 \cdot x$

Step 6.2.3.2

Rewrite $- 1 \cdot x$ as $- x$ .

$y = - x$

Step 7

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

$f^{- 1} (x) = - x$

Step 8

Verify if $f^{- 1} (x) = - x$ is the inverse of $f (x) = - x$ .

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Step 8.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 8.2

Evaluate $f^{- 1} (f (x))$ .

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Step 8.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 8.2.2

Evaluate $f^{- 1} (- x)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- x) = - (- x)$

Step 8.2.3

Multiply $- (- x)$ .

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Step 8.2.3.1

Multiply $- 1$ by $- 1$ .

$f^{- 1} (- x) = 1 x$

Step 8.2.3.2

Multiply $x$ by $1$ .

$f^{- 1} (- x) = x$

Step 8.3

Evaluate $f (f^{- 1} (x))$ .

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Step 8.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 8.3.2

Evaluate $f (- x)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- x) = - (- x)$

Step 8.3.3

Multiply $- (- x)$ .

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Step 8.3.3.1

Multiply $- 1$ by $- 1$ .

$f (- x) = 1 x$

Step 8.3.3.2

Multiply $x$ by $1$ .

$f (- x) = x$

Step 8.4

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

$f^{- 1} (x) = - x$