Precalculus Examples

f (x) = - 8 x - 7

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

Step 1

Write

f (x) = - 8 x - 7

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

y = - 8 x - 7

$y = - 8 x - 7$

Step 2

Interchange the variables.

x = - 8 y - 7

$x = - 8 y - 7$

Step 3

Solve for

y

$y$ .

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Step 3.1

Rewrite the equation as

- 8 y - 7 = x

$- 8 y - 7 = x$ .

- 8 y - 7 = x

$- 8 y - 7 = x$

Step 3.2

Add

7

$7$ to both sides of the equation.

- 8 y = x + 7

$- 8 y = x + 7$

Step 3.3

Divide each term in

- 8 y = x + 7

$- 8 y = x + 7$ by

- 8

$- 8$ and simplify.

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Step 3.3.1

Divide each term in

- 8 y = x + 7

$- 8 y = x + 7$ by

- 8

$- 8$ .

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

$- 8$ .

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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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