Finite Math Examples

$e^{2 x}$

Step 1

Interchange the variables.

$x = e^{2 y}$

Step 2

Solve for $y$ .

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Step 2.1

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

$e^{2 y} = x$

Step 2.2

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

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

Step 2.3

Expand the left side.

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Step 2.3.1

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

$2 y \ln (e) = \ln (x)$

Step 2.3.2

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

$2 y \cdot 1 = \ln (x)$

Step 2.3.3

Multiply $2$ by $1$ .

$2 y = \ln (x)$

Step 2.4

Divide each term in $2 y = \ln (x)$ by $2$ and simplify.

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Step 2.4.1

Divide each term in $2 y = \ln (x)$ by $2$ .

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

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $2$ .

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Step 2.4.2.1.1

Cancel the common factor.

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

Step 2.4.2.1.2

Divide $y$ by $1$ .

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

Step 3

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

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

Step 4

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

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.2.4

Cancel the common factor of $2$ .

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Step 4.2.4.1

Cancel the common factor.

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

Step 4.2.4.2

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

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

Step 4.2.5

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

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

Step 4.2.6

Multiply $x$ by $1$ .

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

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

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

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

Step 4.3.3

Cancel the common factor of $2$ .

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Step 4.3.3.1

Cancel the common factor.

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

Step 4.3.3.2

Rewrite the expression.

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

Step 4.3.4

Exponentiation and log are inverse functions.

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

Step 4.4

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

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