Precalculus Examples

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

Step 1

Write $f (x) = \frac{x - 7}{7}$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{y - 7}{7} = x$ .

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

Step 3.2

Multiply both sides of the equation by $7$ .

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

Step 3.3

Simplify the left side.

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Step 3.3.1

Cancel the common factor of $7$ .

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Step 3.3.1.1

Cancel the common factor.

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

Step 3.3.1.2

Rewrite the expression.

$y - 7 = 7 x$

Step 3.4

Add $7$ to both sides of the equation.

$y = 7 x + 7$

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Cancel the common factor of $7$ .

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Step 5.2.3.1

Cancel the common factor.

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

Step 5.2.3.2

Rewrite the expression.

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

Step 5.2.4

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

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Step 5.2.4.1

Add $- 7$ and $7$ .

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

Step 5.2.4.2

Add $x$ and $0$ .

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

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

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

Step 5.3.3

Cancel the common factor of $(7 x + 7) - 7$ and $7$ .

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Step 5.3.3.1

Factor $7$ out of $7 x$ .

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

Step 5.3.3.2

Factor $7$ out of $7$ .

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

Step 5.3.3.3

Factor $7$ out of $7 (x) + 7 (1)$ .

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

Step 5.3.3.4

Factor $7$ out of $- 7$ .

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

Step 5.3.3.5

Factor $7$ out of $7 (x + 1) + 7 (- 1)$ .

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

Step 5.3.3.6

Cancel the common factors.

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Step 5.3.3.6.1

Factor $7$ out of $7$ .

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

Step 5.3.3.6.2

Cancel the common factor.

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

Step 5.3.3.6.3

Rewrite the expression.

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

Step 5.3.3.6.4

Divide $x + 1 - 1$ by $1$ .

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

Step 5.3.4

Combine the opposite terms in $x + 1 - 1$ .

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Step 5.3.4.1

Subtract $1$ from $1$ .

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

Step 5.3.4.2

Add $x$ and $0$ .

$f (7 x + 7) = x$

Step 5.4

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

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