Find the Inverse g(x)

$g (x)$

Step 1

Interchange the variables.

$g = y (x)$

Step 2

Solve for $y$ .

Rewrite the equation as $y x = g$ .

$y x = g$

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

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

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

Simplify the left side.

Cancel the common factor of $x$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Step 3

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

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

Step 4

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

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

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

Set up the composite result function.

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

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

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

Cancel the common factor of $x$ .

Cancel the common factor.

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

Divide $g$ by $1$ .

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

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

Set up the composite result function.

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

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

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

Cancel the common factor of $x$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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