Find the Inverse y=e^(x/2)

$y = e^{\frac{x}{2}}$

Interchange the variables.

$x = e^{\frac{y}{2}}$

Solve for $y$ .

Rewrite the equation as $e^{\frac{y}{2}} = x$ .

$e^{\frac{y}{2}} = x$

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{\frac{y}{2}}) = \ln (x)$

Expand the left side.

Expand $\ln (e^{\frac{y}{2}})$ by moving $\frac{y}{2}$ outside the logarithm.

$\frac{y}{2} \ln (e) = \ln (x)$

The natural logarithm of $e$ is $1$ .

$\frac{y}{2} \cdot 1 = \ln (x)$

Multiply $\frac{y}{2}$ by $1$ .

$\frac{y}{2} = \ln (x)$

Multiply both sides of the equation by $2$ .

$2 \frac{y}{2} = 2 \ln (x)$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$2 \frac{y}{2} = 2 \ln (x)$

Rewrite the expression.

$y = 2 \ln (x)$

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

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

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

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

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

Use logarithm rules to move $\frac{x}{2}$ out of the exponent.

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

The natural logarithm of $e$ is $1$ .

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

Multiply $\frac{x}{2}$ by $1$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

Set up the composite result function.

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

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

$f (2 \ln (x)) = e^{\frac{2 \ln (x)}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (2 \ln (x)) = e^{\frac{2 \ln (x)}{2}}$

Divide $\ln (x)$ by $1$ .

$f (2 \ln (x)) = e^{\ln (x)}$

Exponentiation and log are inverse functions.

$f (2 \ln (x)) = x$

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

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

Find the Inverse y=e^(x/2)

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