Precalculus Examples

$f (x) = 3 e^{x}$

Step 1

Write $f (x) = 3 e^{x}$ as an equation.

$y = 3 e^{x}$

Step 2

Interchange the variables.

$x = 3 e^{y}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $3 e^{y} = x$ .

$3 e^{y} = x$

Step 3.2

Divide each term in $3 e^{y} = x$ by $3$ and simplify.

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Step 3.2.1

Divide each term in $3 e^{y} = x$ by $3$ .

$\frac{3 e^{y}}{3} = \frac{x}{3}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{3 e^{y}}{3} = \frac{x}{3}$

Step 3.2.2.1.2

Divide $e^{y}$ by $1$ .

$e^{y} = \frac{x}{3}$

Step 3.3

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

$\ln (e^{y}) = \ln (\frac{x}{3})$

Step 3.4

Expand the left side.

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Step 3.4.1

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

$y \ln (e) = \ln (\frac{x}{3})$

Step 3.4.2

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

$y \cdot 1 = \ln (\frac{x}{3})$

Step 3.4.3

Multiply $y$ by $1$ .

$y = \ln (\frac{x}{3})$

Step 4

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

$f^{- 1} (x) = \ln (\frac{x}{3})$

Step 5

Verify if $f^{- 1} (x) = \ln (\frac{x}{3})$ is the inverse of $f (x) = 3 e^{x}$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (3 e^{x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (3 e^{x}) = \ln (\frac{3 e^{x}}{3})$

Step 5.2.3

Cancel the common factor of $3$ .

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Step 5.2.3.1

Cancel the common factor.

$f^{- 1} (3 e^{x}) = \ln (\frac{3 e^{x}}{3})$

Step 5.2.3.2

Divide $e^{x}$ by $1$ .

$f^{- 1} (3 e^{x}) = \ln (e^{x})$

Step 5.2.4

Use logarithm rules to move $x$ out of the exponent.

$f^{- 1} (3 e^{x}) = x \ln (e)$

Step 5.2.5

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

$f^{- 1} (3 e^{x}) = x \cdot 1$

Step 5.2.6

Multiply $x$ by $1$ .

$f^{- 1} (3 e^{x}) = x$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (\ln (\frac{x}{3}))$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\ln (\frac{x}{3})) = 3 e^{\ln (\frac{x}{3})}$

Step 5.3.3

Exponentiation and log are inverse functions.

$f (\ln (\frac{x}{3})) = 3 (\frac{x}{3})$

Step 5.3.4

Cancel the common factor of $3$ .

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Step 5.3.4.1

Cancel the common factor.

$f (\ln (\frac{x}{3})) = 3 (\frac{x}{3})$

Step 5.3.4.2

Rewrite the expression.

$f (\ln (\frac{x}{3})) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \ln (\frac{x}{3})$ is the inverse of $f (x) = 3 e^{x}$ .

$f^{- 1} (x) = \ln (\frac{x}{3})$