Finite Math Examples

$f (x) = - x - 7$

Step 1

Write $f (x) = - x - 7$ as an equation.

$y = - x - 7$

Step 2

Interchange the variables.

$x = - y - 7$

Step 3

Solve for $y$ .

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Rewrite the equation as $- y - 7 = x$ .

$- y - 7 = x$

Add $7$ to both sides of the equation.

$- y = x + 7$

Divide each term in $- y = x + 7$ by $- 1$ and simplify.

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Divide each term in $- y = x + 7$ by $- 1$ .

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

Simplify the left side.

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Dividing two negative values results in a positive value.

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

Divide $y$ by $1$ .

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

Simplify the right side.

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Simplify each term.

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

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

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

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

Divide $7$ by $- 1$ .

$y = - x - 7$

Step 4

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

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

Step 5

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

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

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

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Set up the composite result function.

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

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

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

Simplify each term.

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Apply the distributive property.

$f^{- 1} (- x - 7) = x + 7 - 7$

Multiply $- - x$ .

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Multiply $- 1$ by $- 1$ .

$f^{- 1} (- x - 7) = 1 x + 7 - 7$

Multiply $x$ by $1$ .

$f^{- 1} (- x - 7) = x + 7 - 7$

Multiply $- 1$ by $- 7$ .

$f^{- 1} (- x - 7) = x + 7 - 7$

Combine the opposite terms in $x + 7 - 7$ .

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Subtract $7$ from $7$ .

$f^{- 1} (- x - 7) = x + 0$

Add $x$ and $0$ .

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

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

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Set up the composite result function.

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

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

$f (- x - 7) = - (- x - 7) - 7$

Simplify each term.

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Apply the distributive property.

$f (- x - 7) = x + 7 - 7$

Multiply $- - x$ .

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Multiply $- 1$ by $- 1$ .

$f (- x - 7) = 1 x + 7 - 7$

Multiply $x$ by $1$ .

$f (- x - 7) = x + 7 - 7$

Multiply $- 1$ by $- 7$ .

$f (- x - 7) = x + 7 - 7$

Combine the opposite terms in $x + 7 - 7$ .

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Subtract $7$ from $7$ .

$f (- x - 7) = x + 0$

Add $x$ and $0$ .

$f (- x - 7) = x$

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

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