Calculus Examples

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

Step 1

Write $f (x) = 3 e^{\frac{1}{3} x + 1}$ as an equation.

$y = 3 e^{\frac{1}{3} x + 1}$

Step 2

Interchange the variables.

$x = 3 e^{\frac{1}{3} y + 1}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $3 e^{\frac{1}{3} y + 1} = x$ .

$3 e^{\frac{1}{3} y + 1} = x$

Step 3.2

Divide each term in $3 e^{\frac{1}{3} y + 1} = x$ by $3$ and simplify.

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Step 3.2.1

Divide each term in $3 e^{\frac{1}{3} y + 1} = x$ by $3$ .

$\frac{3 e^{\frac{1}{3} y + 1}}{3} = \frac{x}{3}$

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.

$\frac{3 e^{\frac{1}{3} y + 1}}{3} = \frac{x}{3}$

Step 3.2.2.1.2

Divide $e^{\frac{1}{3} y + 1}$ by $1$ .

$e^{\frac{1}{3} y + 1} = \frac{x}{3}$

Step 3.2.2.2

Combine $\frac{1}{3}$ and $y$ .

$e^{\frac{y}{3} + 1} = \frac{x}{3}$

Step 3.3

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

$\ln (e^{\frac{y}{3} + 1}) = \ln (\frac{x}{3})$

Step 3.4

Expand the left side.

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Step 3.4.1

Expand $\ln (e^{\frac{y}{3} + 1})$ by moving $\frac{y}{3} + 1$ outside the logarithm.

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

Step 3.4.2

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

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

Step 3.4.3

Multiply $\frac{y}{3} + 1$ by $1$ .

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

Step 3.5

Subtract $1$ from both sides of the equation.

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

Step 3.6

Multiply both sides of the equation by $3$ .

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

Step 3.7

Simplify both sides of the equation.

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Step 3.7.1

Simplify the left side.

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Step 3.7.1.1

Cancel the common factor of $3$ .

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Step 3.7.1.1.1

Cancel the common factor.

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

Step 3.7.1.1.2

Rewrite the expression.

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

Step 3.7.2

Simplify the right side.

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Step 3.7.2.1

Simplify $3 (\ln (\frac{x}{3}) - 1)$ .

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Step 3.7.2.1.1

Apply the distributive property.

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

Step 3.7.2.1.2

Multiply $3$ by $- 1$ .

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Cancel the common factor of $3$ .

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Step 5.2.3.1.1

Cancel the common factor.

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

Step 5.2.3.1.2

Divide $e^{\frac{1}{3} x + 1}$ by $1$ .

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

Step 5.2.3.2

Use logarithm rules to move $\frac{1}{3} x + 1$ out of the exponent.

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

Step 5.2.3.3

Combine $\frac{1}{3}$ and $x$ .

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

Step 5.2.3.4

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

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

Step 5.2.3.5

Multiply $\frac{x}{3} + 1$ by $1$ .

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

Step 5.2.3.6

Apply the distributive property.

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

Step 5.2.3.7

Cancel the common factor of $3$ .

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Step 5.2.3.7.1

Cancel the common factor.

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

Step 5.2.3.7.2

Rewrite the expression.

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

Step 5.2.3.8

Multiply $3$ by $1$ .

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

Step 5.2.4

Combine the opposite terms in $x + 3 - 3$ .

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Step 5.2.4.1

Subtract $3$ from $3$ .

$f^{- 1} (3 e^{\frac{1}{3} x + 1}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

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

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

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

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Simplify $3 \ln (\frac{x}{3})$ by moving $3$ inside the logarithm.

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

Step 5.3.3.1.2

Apply the product rule to $\frac{x}{3}$ .

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

Step 5.3.3.1.3

Raise $3$ to the power of $3$ .

$f (3 \ln (\frac{x}{3}) - 3) = 3 e^{\frac{1}{3} \cdot (\ln (\frac{x^{3}}{27}) - 3) + 1}$

Step 5.3.3.2

Apply the distributive property.

$f (3 \ln (\frac{x}{3}) - 3) = 3 e^{\frac{1}{3} \cdot \ln (\frac{x^{3}}{27}) + \frac{1}{3} \cdot - 3 + 1}$

Step 5.3.3.3

Simplify $\frac{1}{3} \ln (\frac{x^{3}}{27})$ by moving $\frac{1}{3}$ inside the logarithm.

$f (3 \ln (\frac{x}{3}) - 3) = 3 e^{\ln ({(\frac{x^{3}}{27})}^{\frac{1}{3}}) + \frac{1}{3} \cdot - 3 + 1}$

Step 5.3.3.4

Cancel the common factor of $3$ .

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Step 5.3.3.4.1

Factor $3$ out of $- 3$ .

$f (3 \ln (\frac{x}{3}) - 3) = 3 e^{\ln ({(\frac{x^{3}}{27})}^{\frac{1}{3}}) + \frac{1}{3} \cdot (3 (- 1)) + 1}$

Step 5.3.3.4.2

Cancel the common factor.

$f (3 \ln (\frac{x}{3}) - 3) = 3 e^{\ln ({(\frac{x^{3}}{27})}^{\frac{1}{3}}) + \frac{1}{3} \cdot (3 \cdot - 1) + 1}$

Step 5.3.3.4.3

Rewrite the expression.

$f (3 \ln (\frac{x}{3}) - 3) = 3 e^{\ln ({(\frac{x^{3}}{27})}^{\frac{1}{3}}) - 1 + 1}$

Step 5.3.3.5

Simplify each term.

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Step 5.3.3.5.1

Apply the product rule to $\frac{x^{3}}{27}$ .

$f (3 \ln (\frac{x}{3}) - 3) = 3 e^{\ln (\frac{{(x^{3})}^{\frac{1}{3}}}{27^{\frac{1}{3}}}) - 1 + 1}$

Step 5.3.3.5.2

Simplify the numerator.

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Step 5.3.3.5.2.1

Multiply the exponents in ${(x^{3})}^{\frac{1}{3}}$ .

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Step 5.3.3.5.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (3 \ln (\frac{x}{3}) - 3) = 3 e^{\ln (\frac{x^{3 (\frac{1}{3})}}{27^{\frac{1}{3}}}) - 1 + 1}$

Step 5.3.3.5.2.1.2

Cancel the common factor of $3$ .

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Step 5.3.3.5.2.1.2.1

Cancel the common factor.

$f (3 \ln (\frac{x}{3}) - 3) = 3 e^{\ln (\frac{x^{3 (\frac{1}{3})}}{27^{\frac{1}{3}}}) - 1 + 1}$

Step 5.3.3.5.2.1.2.2

Rewrite the expression.

$f (3 \ln (\frac{x}{3}) - 3) = 3 e^{\ln (\frac{x^{1}}{27^{\frac{1}{3}}}) - 1 + 1}$

Step 5.3.3.5.2.2

Simplify.

$f (3 \ln (\frac{x}{3}) - 3) = 3 e^{\ln (\frac{x}{27^{\frac{1}{3}}}) - 1 + 1}$

Step 5.3.3.5.3

Simplify the denominator.

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Step 5.3.3.5.3.1

Rewrite $27$ as $3^{3}$ .

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

Step 5.3.3.5.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 5.3.3.5.3.3

Cancel the common factor of $3$ .

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Step 5.3.3.5.3.3.1

Cancel the common factor.

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

Step 5.3.3.5.3.3.2

Rewrite the expression.

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

Step 5.3.3.5.3.4

Evaluate the exponent.

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

Step 5.3.4

Combine the opposite terms in $\ln (\frac{x}{3}) - 1 + 1$ .

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Step 5.3.4.1

Add $- 1$ and $1$ .

$f (3 \ln (\frac{x}{3}) - 3) = 3 e^{\ln (\frac{x}{3}) + 0}$

Step 5.3.4.2

Add $\ln (\frac{x}{3})$ and $0$ .

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

Step 5.3.5

Exponentiation and log are inverse functions.

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

Step 5.3.6

Cancel the common factor of $3$ .

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Step 5.3.6.1

Cancel the common factor.

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

Step 5.3.6.2

Rewrite the expression.

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

Step 5.4

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

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