Find the Inverse (x-2)/7+8

$\frac{x - 2}{7} + 8$

Step 1

Interchange the variables.

$x = \frac{y - 2}{7} + 8$

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{y - 2}{7} + 8 = x$ .

$\frac{y - 2}{7} + 8 = x$

Simplify $\frac{y - 2}{7} + 8$ .

To write $8$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{y - 2}{7} + 8 \cdot \frac{7}{7} = x$

Simplify terms.

Combine $8$ and $\frac{7}{7}$ .

$\frac{y - 2}{7} + \frac{8 \cdot 7}{7} = x$

Combine the numerators over the common denominator.

$\frac{y - 2 + 8 \cdot 7}{7} = x$

Simplify the numerator.

Multiply $8$ by $7$ .

$\frac{y - 2 + 56}{7} = x$

Add $- 2$ and $56$ .

$\frac{y + 54}{7} = x$

Multiply both sides of the equation by $7$ .

$7 \frac{y + 54}{7} = 7 x$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$7 \frac{y + 54}{7} = 7 x$

Rewrite the expression.

$y + 54 = 7 x$

Subtract $54$ from both sides of the equation.

$y = 7 x - 54$

Step 3

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

$f^{- 1} (x) = 7 x - 54$

Step 4

Verify if $f^{- 1} (x) = 7 x - 54$ is the inverse of $f (x) = \frac{x - 2}{7} + 8$ .

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

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

Set up the composite result function.

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

Evaluate $f^{- 1} (\frac{x - 2}{7} + 8)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x - 2}{7} + 8) = 7 (\frac{x - 2}{7} + 8) - 54$

Simplify each term.

To write $8$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$f^{- 1} (\frac{x - 2}{7} + 8) = 7 (\frac{x - 2}{7} + 8 \cdot \frac{7}{7}) - 54$

Combine $8$ and $\frac{7}{7}$ .

$f^{- 1} (\frac{x - 2}{7} + 8) = 7 (\frac{x - 2}{7} + \frac{8 \cdot 7}{7}) - 54$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{x - 2}{7} + 8) = 7 (\frac{x - 2 + 8 \cdot 7}{7}) - 54$

Simplify the numerator.

Multiply $8$ by $7$ .

$f^{- 1} (\frac{x - 2}{7} + 8) = 7 (\frac{x - 2 + 56}{7}) - 54$

Add $- 2$ and $56$ .

$f^{- 1} (\frac{x - 2}{7} + 8) = 7 (\frac{x + 54}{7}) - 54$

Cancel the common factor of $7$ .

Cancel the common factor.

$f^{- 1} (\frac{x - 2}{7} + 8) = 7 (\frac{x + 54}{7}) - 54$

Rewrite the expression.

$f^{- 1} (\frac{x - 2}{7} + 8) = x + 54 - 54$

Combine the opposite terms in $x + 54 - 54$ .

Subtract $54$ from $54$ .

$f^{- 1} (\frac{x - 2}{7} + 8) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (\frac{x - 2}{7} + 8) = x$

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

Set up the composite result function.

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

Evaluate $f (7 x - 54)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (7 x - 54) = \frac{(7 x - 54) - 2}{7} + 8$

Subtract $2$ from $- 54$ .

$f (7 x - 54) = \frac{7 x - 56}{7} + 8$

Cancel the common factor of $7 x - 56$ and $7$ .

Factor $7$ out of $7 x$ .

$f (7 x - 54) = \frac{7 (x) - 56}{7} + 8$

Factor $7$ out of $- 56$ .

$f (7 x - 54) = \frac{7 (x) + 7 \cdot - 8}{7} + 8$

Factor $7$ out of $7 (x) + 7 (- 8)$ .

$f (7 x - 54) = \frac{7 (x - 8)}{7} + 8$

Cancel the common factors.

Factor $7$ out of $7$ .

$f (7 x - 54) = \frac{7 (x - 8)}{7 (1)} + 8$

Cancel the common factor.

$f (7 x - 54) = \frac{7 (x - 8)}{7 \cdot 1} + 8$

Rewrite the expression.

$f (7 x - 54) = \frac{x - 8}{1} + 8$

Divide $x - 8$ by $1$ .

$f (7 x - 54) = x - 8 + 8$

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

Add $- 8$ and $8$ .

$f (7 x - 54) = x + 0$

Add $x$ and $0$ .

$f (7 x - 54) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 7 x - 54$ is the inverse of $f (x) = \frac{x - 2}{7} + 8$ .

$f^{- 1} (x) = 7 x - 54$

Find the Inverse (x-2)/7+8

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