Find the Inverse (x-7)/9+10

$\frac{x - 7}{9} + 10$

Step 1

Interchange the variables.

$x = \frac{y - 7}{9} + 10$

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{y - 7}{9} + 10 = x$ .

$\frac{y - 7}{9} + 10 = x$

Simplify $\frac{y - 7}{9} + 10$ .

To write $10$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{y - 7}{9} + 10 \cdot \frac{9}{9} = x$

Simplify terms.

Combine $10$ and $\frac{9}{9}$ .

$\frac{y - 7}{9} + \frac{10 \cdot 9}{9} = x$

Combine the numerators over the common denominator.

$\frac{y - 7 + 10 \cdot 9}{9} = x$

Simplify the numerator.

Multiply $10$ by $9$ .

$\frac{y - 7 + 90}{9} = x$

Add $- 7$ and $90$ .

$\frac{y + 83}{9} = x$

Multiply both sides of the equation by $9$ .

$9 \frac{y + 83}{9} = 9 x$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$9 \frac{y + 83}{9} = 9 x$

Rewrite the expression.

$y + 83 = 9 x$

Subtract $83$ from both sides of the equation.

$y = 9 x - 83$

Step 3

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

$f^{- 1} (x) = 9 x - 83$

Step 4

Verify if $f^{- 1} (x) = 9 x - 83$ is the inverse of $f (x) = \frac{x - 7}{9} + 10$ .

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 $f^{- 1} (\frac{x - 7}{9} + 10)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x - 7}{9} + 10) = 9 (\frac{x - 7}{9} + 10) - 83$

Simplify each term.

To write $10$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f^{- 1} (\frac{x - 7}{9} + 10) = 9 (\frac{x - 7}{9} + 10 \cdot \frac{9}{9}) - 83$

Combine $10$ and $\frac{9}{9}$ .

$f^{- 1} (\frac{x - 7}{9} + 10) = 9 (\frac{x - 7}{9} + \frac{10 \cdot 9}{9}) - 83$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{x - 7}{9} + 10) = 9 (\frac{x - 7 + 10 \cdot 9}{9}) - 83$

Simplify the numerator.

Multiply $10$ by $9$ .

$f^{- 1} (\frac{x - 7}{9} + 10) = 9 (\frac{x - 7 + 90}{9}) - 83$

Add $- 7$ and $90$ .

$f^{- 1} (\frac{x - 7}{9} + 10) = 9 (\frac{x + 83}{9}) - 83$

Cancel the common factor of $9$ .

Cancel the common factor.

$f^{- 1} (\frac{x - 7}{9} + 10) = 9 (\frac{x + 83}{9}) - 83$

Rewrite the expression.

$f^{- 1} (\frac{x - 7}{9} + 10) = x + 83 - 83$

Combine the opposite terms in $x + 83 - 83$ .

Subtract $83$ from $83$ .

$f^{- 1} (\frac{x - 7}{9} + 10) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (\frac{x - 7}{9} + 10) = x$

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

Set up the composite result function.

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

Evaluate $f (9 x - 83)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (9 x - 83) = \frac{(9 x - 83) - 7}{9} + 10$

Subtract $7$ from $- 83$ .

$f (9 x - 83) = \frac{9 x - 90}{9} + 10$

Cancel the common factor of $9 x - 90$ and $9$ .

Factor $9$ out of $9 x$ .

$f (9 x - 83) = \frac{9 (x) - 90}{9} + 10$

Factor $9$ out of $- 90$ .

$f (9 x - 83) = \frac{9 (x) + 9 \cdot - 10}{9} + 10$

Factor $9$ out of $9 (x) + 9 (- 10)$ .

$f (9 x - 83) = \frac{9 (x - 10)}{9} + 10$

Cancel the common factors.

Factor $9$ out of $9$ .

$f (9 x - 83) = \frac{9 (x - 10)}{9 (1)} + 10$

Cancel the common factor.

$f (9 x - 83) = \frac{9 (x - 10)}{9 \cdot 1} + 10$

Rewrite the expression.

$f (9 x - 83) = \frac{x - 10}{1} + 10$

Divide $x - 10$ by $1$ .

$f (9 x - 83) = x - 10 + 10$

Combine the opposite terms in $x - 10 + 10$ .

Add $- 10$ and $10$ .

$f (9 x - 83) = x + 0$

Add $x$ and $0$ .

$f (9 x - 83) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 9 x - 83$ is the inverse of $f (x) = \frac{x - 7}{9} + 10$ .

$f^{- 1} (x) = 9 x - 83$

Find the Inverse (x-7)/9+10

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