Algebra Examples

$f (x) = 4 + \frac{5}{e^{x - 3}}$

Step 1

Write $f (x) = 4 + \frac{5}{e^{x - 3}}$ as an equation.

$y = 4 + \frac{5}{e^{x - 3}}$

Step 2

Interchange the variables.

$x = 4 + \frac{5}{e^{y - 3}}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $4 + \frac{5}{e^{y - 3}} = x$ .

$4 + \frac{5}{e^{y - 3}} = x$

Step 3.2

Subtract $4$ from both sides of the equation.

$\frac{5}{e^{y - 3}} = x - 4$

Step 3.3

Multiply both sides by $e^{y - 3}$ .

$\frac{5}{e^{y - 3}} e^{y - 3} = (x - 4) e^{y - 3}$

Step 3.4

Simplify.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

Cancel the common factor of $e^{y - 3}$ .

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Step 3.4.1.1.1

Cancel the common factor.

$\frac{5}{e^{y - 3}} e^{y - 3} = (x - 4) e^{y - 3}$

Step 3.4.1.1.2

Rewrite the expression.

$5 = (x - 4) e^{y - 3}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Apply the distributive property.

$5 = x e^{y - 3} - 4 e^{y - 3}$

Step 3.5

Solve for $y$ .

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Step 3.5.1

Rewrite the equation as $x e^{y - 3} - 4 e^{y - 3} = 5$ .

$x e^{y - 3} - 4 e^{y - 3} = 5$

Step 3.5.2

Factor $e^{y - 3}$ out of $x e^{y - 3} - 4 e^{y - 3}$ .

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Step 3.5.2.1

Factor $e^{y - 3}$ out of $x e^{y - 3}$ .

$e^{y - 3} x - 4 e^{y - 3} = 5$

Step 3.5.2.2

Factor $e^{y - 3}$ out of $- 4 e^{y - 3}$ .

$e^{y - 3} x + e^{y - 3} \cdot - 4 = 5$

Step 3.5.2.3

Factor $e^{y - 3}$ out of $e^{y - 3} x + e^{y - 3} \cdot - 4$ .

$e^{y - 3} (x - 4) = 5$

Step 3.5.3

Divide each term in $e^{y - 3} (x - 4) = 5$ by $x - 4$ and simplify.

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Step 3.5.3.1

Divide each term in $e^{y - 3} (x - 4) = 5$ by $x - 4$ .

$\frac{e^{y - 3} (x - 4)}{x - 4} = \frac{5}{x - 4}$

Step 3.5.3.2

Simplify the left side.

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Step 3.5.3.2.1

Cancel the common factor of $x - 4$ .

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Step 3.5.3.2.1.1

Cancel the common factor.

$\frac{e^{y - 3} (x - 4)}{x - 4} = \frac{5}{x - 4}$

Step 3.5.3.2.1.2

Divide $e^{y - 3}$ by $1$ .

$e^{y - 3} = \frac{5}{x - 4}$

Step 3.5.4

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

$\ln (e^{y - 3}) = \ln (\frac{5}{x - 4})$

Step 3.5.5

Expand the left side.

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Step 3.5.5.1

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

$(y - 3) \ln (e) = \ln (\frac{5}{x - 4})$

Step 3.5.5.2

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

$(y - 3) \cdot 1 = \ln (\frac{5}{x - 4})$

Step 3.5.5.3

Multiply $y - 3$ by $1$ .

$y - 3 = \ln (\frac{5}{x - 4})$

Step 3.5.6

Add $3$ to both sides of the equation.

$y = \ln (\frac{5}{x - 4}) + 3$

Step 4

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

$f^{- 1} (x) = \ln (\frac{5}{x - 4}) + 3$

Step 5

Verify if $f^{- 1} (x) = \ln (\frac{5}{x - 4}) + 3$ is the inverse of $f (x) = 4 + \frac{5}{e^{x - 3}}$ .

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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 $f^{- 1} (4 + \frac{5}{e^{x - 3}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (4 + \frac{5}{e^{x - 3}}) = \ln (\frac{5}{(4 + \frac{5}{e^{x - 3}}) - 4}) + 3$

Step 5.2.3

Combine the opposite terms in $\ln (\frac{5}{(4 + \frac{5}{e^{x - 3}}) - 4}) + 3$ .

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Step 5.2.3.1

Subtract $4$ from $4$ .

$f^{- 1} (4 + \frac{5}{e^{x - 3}}) = \ln (\frac{5}{0 + \frac{5}{e^{x - 3}}}) + 3$

Step 5.2.3.2

Add $0$ and $\frac{5}{e^{x - 3}}$ .

$f^{- 1} (4 + \frac{5}{e^{x - 3}}) = \ln (\frac{5}{\frac{5}{e^{x - 3}}}) + 3$

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (4 + \frac{5}{e^{x - 3}}) = \ln (5 (\frac{e^{x - 3}}{5})) + 3$

Step 5.2.4.2

Cancel the common factor of $5$ .

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Step 5.2.4.2.1

Cancel the common factor.

$f^{- 1} (4 + \frac{5}{e^{x - 3}}) = \ln (5 (\frac{e^{x - 3}}{5})) + 3$

Step 5.2.4.2.2

Rewrite the expression.

$f^{- 1} (4 + \frac{5}{e^{x - 3}}) = \ln (e^{x - 3}) + 3$

Step 5.2.4.3

Use logarithm rules to move $x - 3$ out of the exponent.

$f^{- 1} (4 + \frac{5}{e^{x - 3}}) = (x - 3) \ln (e) + 3$

Step 5.2.4.4

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

$f^{- 1} (4 + \frac{5}{e^{x - 3}}) = (x - 3) \cdot 1 + 3$

Step 5.2.4.5

Multiply $x - 3$ by $1$ .

$f^{- 1} (4 + \frac{5}{e^{x - 3}}) = x - 3 + 3$

Step 5.2.5

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

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Step 5.2.5.1

Add $- 3$ and $3$ .

$f^{- 1} (4 + \frac{5}{e^{x - 3}}) = x + 0$

Step 5.2.5.2

Add $x$ and $0$ .

$f^{- 1} (4 + \frac{5}{e^{x - 3}}) = x$

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 $f (\ln (\frac{5}{x - 4}) + 3)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\ln (\frac{5}{x - 4}) + 3) = 4 + \frac{5}{e^{(\ln (\frac{5}{x - 4}) + 3) - 3}}$

Step 5.3.3

Combine the opposite terms in $4 + \frac{5}{e^{(\ln (\frac{5}{x - 4}) + 3) - 3}}$ .

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Step 5.3.3.1

Subtract $3$ from $3$ .

$f (\ln (\frac{5}{x - 4}) + 3) = 4 + \frac{5}{e^{\ln (\frac{5}{x - 4}) + 0}}$

Step 5.3.3.2

Add $\ln (\frac{5}{x - 4})$ and $0$ .

$f (\ln (\frac{5}{x - 4}) + 3) = 4 + \frac{5}{e^{\ln (\frac{5}{x - 4})}}$

Step 5.3.4

Simplify each term.

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Step 5.3.4.1

Exponentiation and log are inverse functions.

$f (\ln (\frac{5}{x - 4}) + 3) = 4 + \frac{5}{\frac{5}{x - 4}}$

Step 5.3.4.2

Multiply the numerator by the reciprocal of the denominator.

$f (\ln (\frac{5}{x - 4}) + 3) = 4 + 5 (\frac{x - 4}{5})$

Step 5.3.4.3

Cancel the common factor of $5$ .

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Step 5.3.4.3.1

Cancel the common factor.

$f (\ln (\frac{5}{x - 4}) + 3) = 4 + 5 (\frac{x - 4}{5})$

Step 5.3.4.3.2

Rewrite the expression.

$f (\ln (\frac{5}{x - 4}) + 3) = 4 + x - 4$

Step 5.3.5

Combine the opposite terms in $4 + x - 4$ .

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Step 5.3.5.1

Subtract $4$ from $4$ .

$f (\ln (\frac{5}{x - 4}) + 3) = x + 0$

Step 5.3.5.2

Add $x$ and $0$ .

$f (\ln (\frac{5}{x - 4}) + 3) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \ln (\frac{5}{x - 4}) + 3$ is the inverse of $f (x) = 4 + \frac{5}{e^{x - 3}}$ .

$f^{- 1} (x) = \ln (\frac{5}{x - 4}) + 3$