Find the Inverse

$f (x) = - 8 x - 7$

Step 1

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

$y = - 8 x - 7$

Step 2

Interchange the variables.

$x = - 8 y - 7$

Step 3

Solve for $y$ .

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Step 3.1

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

$- 8 y - 7 = x$

Step 3.2

Add $7$ to both sides of the equation.

$- 8 y = x + 7$

Step 3.3

Divide each term in $- 8 y = x + 7$ by $- 8$ and simplify.

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Step 3.3.1

Divide each term in $- 8 y = x + 7$ by $- 8$ .

$\frac{- 8 y}{- 8} = \frac{x}{- 8} + \frac{7}{- 8}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 8$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 8 y}{- 8} = \frac{x}{- 8} + \frac{7}{- 8}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{x}{- 8} + \frac{7}{- 8}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Move the negative in front of the fraction.

$y = - \frac{x}{8} + \frac{7}{- 8}$

Step 3.3.3.1.2

Move the negative in front of the fraction.

$y = - \frac{x}{8} - \frac{7}{8}$

Step 4

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

$f^{- 1} (x) = - \frac{x}{8} - \frac{7}{8}$

Step 5

Verify if $f^{- 1} (x) = - \frac{x}{8} - \frac{7}{8}$ is the inverse of $f (x) = - 8 x - 7$ .

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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 $f^{- 1} (- 8 x - 7)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- 8 x - 7) = - \frac{- 8 x - 7}{8} - \frac{7}{8}$

Step 5.2.3

Combine the numerators over the common denominator.

$f^{- 1} (- 8 x - 7) = \frac{- (- 8 x - 7) - 7}{8}$

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

Apply the distributive property.

$f^{- 1} (- 8 x - 7) = \frac{- (- 8 x) + 7 - 7}{8}$

Step 5.2.4.2

Multiply $- 8$ by $- 1$ .

$f^{- 1} (- 8 x - 7) = \frac{8 x + 7 - 7}{8}$

Step 5.2.4.3

Multiply $- 1$ by $- 7$ .

$f^{- 1} (- 8 x - 7) = \frac{8 x + 7 - 7}{8}$

Step 5.2.5

Simplify terms.

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Step 5.2.5.1

Combine the opposite terms in $8 x + 7 - 7$ .

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Step 5.2.5.1.1

Subtract $7$ from $7$ .

$f^{- 1} (- 8 x - 7) = \frac{8 x + 0}{8}$

Step 5.2.5.1.2

Add $8 x$ and $0$ .

$f^{- 1} (- 8 x - 7) = \frac{8 x}{8}$

Step 5.2.5.2

Cancel the common factor of $8$ .

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Step 5.2.5.2.1

Cancel the common factor.

$f^{- 1} (- 8 x - 7) = \frac{8 x}{8}$

Step 5.2.5.2.2

Divide $x$ by $1$ .

$f^{- 1} (- 8 x - 7) = x$

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 $f (- \frac{x}{8} - \frac{7}{8})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{x}{8} - \frac{7}{8}) = - 8 (- \frac{x}{8} - \frac{7}{8}) - 7$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$f (- \frac{x}{8} - \frac{7}{8}) = - 8 (- \frac{x}{8}) - 8 (- \frac{7}{8}) - 7$

Step 5.3.3.2

Cancel the common factor of $8$ .

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Step 5.3.3.2.1

Move the leading negative in $- \frac{x}{8}$ into the numerator.

$f (- \frac{x}{8} - \frac{7}{8}) = - 8 \frac{- x}{8} - 8 (- \frac{7}{8}) - 7$

Step 5.3.3.2.2

Factor $8$ out of $- 8$ .

$f (- \frac{x}{8} - \frac{7}{8}) = 8 (- 1) (\frac{- x}{8}) - 8 (- \frac{7}{8}) - 7$

Step 5.3.3.2.3

Cancel the common factor.

$f (- \frac{x}{8} - \frac{7}{8}) = 8 \cdot (- 1 \frac{- x}{8}) - 8 (- \frac{7}{8}) - 7$

Step 5.3.3.2.4

Rewrite the expression.

$f (- \frac{x}{8} - \frac{7}{8}) = - 1 (- x) - 8 (- \frac{7}{8}) - 7$

Step 5.3.3.3

Multiply $- 1$ by $- 1$ .

$f (- \frac{x}{8} - \frac{7}{8}) = 1 x - 8 (- \frac{7}{8}) - 7$

Step 5.3.3.4

Multiply $x$ by $1$ .

$f (- \frac{x}{8} - \frac{7}{8}) = x - 8 (- \frac{7}{8}) - 7$

Step 5.3.3.5

Cancel the common factor of $8$ .

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Step 5.3.3.5.1

Move the leading negative in $- \frac{7}{8}$ into the numerator.

$f (- \frac{x}{8} - \frac{7}{8}) = x - 8 (\frac{- 7}{8}) - 7$

Step 5.3.3.5.2

Factor $8$ out of $- 8$ .

$f (- \frac{x}{8} - \frac{7}{8}) = x + 8 (- 1) (\frac{- 7}{8}) - 7$

Step 5.3.3.5.3

Cancel the common factor.

$f (- \frac{x}{8} - \frac{7}{8}) = x + 8 \cdot (- 1 (\frac{- 7}{8})) - 7$

Step 5.3.3.5.4

Rewrite the expression.

$f (- \frac{x}{8} - \frac{7}{8}) = x - 1 \cdot - 7 - 7$

Step 5.3.3.6

Multiply $- 1$ by $- 7$ .

$f (- \frac{x}{8} - \frac{7}{8}) = x + 7 - 7$

Step 5.3.4

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

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Step 5.3.4.1

Subtract $7$ from $7$ .

$f (- \frac{x}{8} - \frac{7}{8}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (- \frac{x}{8} - \frac{7}{8}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{x}{8} - \frac{7}{8}$ is the inverse of $f (x) = - 8 x - 7$ .

$f^{- 1} (x) = - \frac{x}{8} - \frac{7}{8}$

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