Find the Inverse f(g(x))

$f (g (x))$

Interchange the variables.

$f = y (g (x))$

Solve for $y$ .

Rewrite the equation as $y g x = f$ .

$y g x = f$

Divide each term in $y g x = f$ by $g x$ and simplify.

Divide each term in $y g x = f$ by $g x$ .

$\frac{y g x}{g x} = \frac{f}{g x}$

Simplify the left side.

Cancel the common factor of $g$ .

Cancel the common factor.

$\frac{y g x}{g x} = \frac{f}{g x}$

Rewrite the expression.

$\frac{y x}{x} = \frac{f}{g x}$

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{y x}{x} = \frac{f}{g x}$

Divide $y$ by $1$ .

$y = \frac{f}{g x}$

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

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

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

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

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

Cancel the common factor of $g$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x$ .

Cancel the common factor.

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

Divide $f$ by $1$ .

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

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

Set up the composite result function.

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

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

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

Cancel the common factor of $g x$ .

Factor $g x$ out of $g (x)$ .

$f (\frac{f}{g x}) = \frac{f}{g x} \cdot (g x (1))$

Cancel the common factor.

$f (\frac{f}{g x}) = \frac{f}{g x} \cdot (g x \cdot 1)$

Rewrite the expression.

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

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

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

Find the Inverse f(g(x))

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