Find the Inverse x/9

$\frac{x}{9}$

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{y}{9} = x$ .

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

Multiply both sides of the equation by $9$ .

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

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

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

Rewrite the expression.

$y = 9 x$

Step 3

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

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

Step 4

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

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

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

Cancel the common factor of $9$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

Set up the composite result function.

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

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

$f (9 x) = \frac{9 x}{9}$

Cancel the common factor of $9$ .

Cancel the common factor.

$f (9 x) = \frac{9 x}{9}$

Divide $x$ by $1$ .

$f (9 x) = x$

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

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