Find the Inverse f(2)

$f (2)$

Step 1

Interchange the variables.

$f = y (2)$

Step 2

Solve for $y$ .

Rewrite the equation as $y (2) = f$ .

$y (2) = f$

Divide each term in $y (2) = f$ by $2$ and simplify.

Divide each term in $y (2) = f$ by $2$ .

$\frac{y (2)}{2} = \frac{f}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{y \cdot 2}{2} = \frac{f}{2}$

Divide $y$ by $1$ .

$y = \frac{f}{2}$

Step 3

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

$f^{- 1} (f) = \frac{f}{2}$

Step 4

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

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

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

Set up the composite result function.

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

Evaluate $f^{- 1} (f (2))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (f (2)) = \frac{f (2)}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (f (2)) = \frac{f \cdot 2}{2}$

Divide $f$ by $1$ .

$f^{- 1} (f (2)) = f$

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

Set up the composite result function.

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

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

$f (\frac{f}{2}) = (\frac{f}{2}) (2)$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (\frac{f}{2}) = \frac{f}{2} \cdot 2$

Rewrite the expression.

$f (\frac{f}{2}) = f$

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

$f^{- 1} (f) = \frac{f}{2}$