Algebra Examples

$f (x) = 3 + \frac{5}{e^{x - 6}}$

Step 1

Write $f (x) = 3 + \frac{5}{e^{x - 6}}$ as an equation.

$y = 3 + \frac{5}{e^{x - 6}}$

Step 2

Interchange the variables.

$x = 3 + \frac{5}{e^{y - 6}}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $3 + \frac{5}{e^{y - 6}} = x$ .

$3 + \frac{5}{e^{y - 6}} = x$

Step 3.2

Subtract $3$ from both sides of the equation.

$\frac{5}{e^{y - 6}} = x - 3$

Step 3.3

Multiply both sides by $e^{y - 6}$ .

$\frac{5}{e^{y - 6}} e^{y - 6} = (x - 3) e^{y - 6}$

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 $e^{y - 6}$ .

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Step 3.4.1.1.1

Cancel the common factor.

$\frac{5}{e^{y - 6}} e^{y - 6} = (x - 3) e^{y - 6}$

Step 3.4.1.1.2

Rewrite the expression.

$5 = (x - 3) e^{y - 6}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Apply the distributive property.

$5 = x e^{y - 6} - 3 e^{y - 6}$

Step 3.5

Solve for $y$ .

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Step 3.5.1

Rewrite the equation as $x e^{y - 6} - 3 e^{y - 6} = 5$ .

$x e^{y - 6} - 3 e^{y - 6} = 5$

Step 3.5.2

Factor $e^{y - 6}$ out of $x e^{y - 6} - 3 e^{y - 6}$ .

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Step 3.5.2.1

Factor $e^{y - 6}$ out of $x e^{y - 6}$ .

$e^{y - 6} x - 3 e^{y - 6} = 5$

Step 3.5.2.2

Factor $e^{y - 6}$ out of $- 3 e^{y - 6}$ .

$e^{y - 6} x + e^{y - 6} \cdot - 3 = 5$

Step 3.5.2.3

Factor $e^{y - 6}$ out of $e^{y - 6} x + e^{y - 6} \cdot - 3$ .

$e^{y - 6} (x - 3) = 5$

Step 3.5.3

Divide each term in $e^{y - 6} (x - 3) = 5$ by $x - 3$ and simplify.

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Step 3.5.3.1

Divide each term in $e^{y - 6} (x - 3) = 5$ by $x - 3$ .

$\frac{e^{y - 6} (x - 3)}{x - 3} = \frac{5}{x - 3}$

Step 3.5.3.2

Simplify the left side.

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Step 3.5.3.2.1

Cancel the common factor of $x - 3$ .

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Step 3.5.3.2.1.1

Cancel the common factor.

$\frac{e^{y - 6} (x - 3)}{x - 3} = \frac{5}{x - 3}$

Step 3.5.3.2.1.2

Divide $e^{y - 6}$ by $1$ .

$e^{y - 6} = \frac{5}{x - 3}$

Step 3.5.4

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

$\ln (e^{y - 6}) = \ln (\frac{5}{x - 3})$

Step 3.5.5

Expand the left side.

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Step 3.5.5.1

Expand $\ln (e^{y - 6})$ by moving $y - 6$ outside the logarithm.

$(y - 6) \ln (e) = \ln (\frac{5}{x - 3})$

Step 3.5.5.2

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

$(y - 6) \cdot 1 = \ln (\frac{5}{x - 3})$

Step 3.5.5.3

Multiply $y - 6$ by $1$ .

$y - 6 = \ln (\frac{5}{x - 3})$

Step 3.5.6

Add $6$ to both sides of the equation.

$y = \ln (\frac{5}{x - 3}) + 6$

Step 4

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

$f^{- 1} (x) = \ln (\frac{5}{x - 3}) + 6$

Step 5

Verify if $f^{- 1} (x) = \ln (\frac{5}{x - 3}) + 6$ is the inverse of $f (x) = 3 + \frac{5}{e^{x - 6}}$ .

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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 $f^{- 1} (3 + \frac{5}{e^{x - 6}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (3 + \frac{5}{e^{x - 6}}) = \ln (\frac{5}{(3 + \frac{5}{e^{x - 6}}) - 3}) + 6$

Step 5.2.3

Combine the opposite terms in $\ln (\frac{5}{(3 + \frac{5}{e^{x - 6}}) - 3}) + 6$ .

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Step 5.2.3.1

Subtract $3$ from $3$ .

$f^{- 1} (3 + \frac{5}{e^{x - 6}}) = \ln (\frac{5}{0 + \frac{5}{e^{x - 6}}}) + 6$

Step 5.2.3.2

Add $0$ and $\frac{5}{e^{x - 6}}$ .

$f^{- 1} (3 + \frac{5}{e^{x - 6}}) = \ln (\frac{5}{\frac{5}{e^{x - 6}}}) + 6$

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.

$f^{- 1} (3 + \frac{5}{e^{x - 6}}) = \ln (5 (\frac{e^{x - 6}}{5})) + 6$

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.

$f^{- 1} (3 + \frac{5}{e^{x - 6}}) = \ln (5 (\frac{e^{x - 6}}{5})) + 6$

Step 5.2.4.2.2

Rewrite the expression.

$f^{- 1} (3 + \frac{5}{e^{x - 6}}) = \ln (e^{x - 6}) + 6$

Step 5.2.4.3

Use logarithm rules to move $x - 6$ out of the exponent.

$f^{- 1} (3 + \frac{5}{e^{x - 6}}) = (x - 6) \ln (e) + 6$

Step 5.2.4.4

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

$f^{- 1} (3 + \frac{5}{e^{x - 6}}) = (x - 6) \cdot 1 + 6$

Step 5.2.4.5

Multiply $x - 6$ by $1$ .

$f^{- 1} (3 + \frac{5}{e^{x - 6}}) = x - 6 + 6$

Step 5.2.5

Combine the opposite terms in $x - 6 + 6$ .

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Step 5.2.5.1

Add $- 6$ and $6$ .

$f^{- 1} (3 + \frac{5}{e^{x - 6}}) = x + 0$

Step 5.2.5.2

Add $x$ and $0$ .

$f^{- 1} (3 + \frac{5}{e^{x - 6}}) = x$

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 $f (\ln (\frac{5}{x - 3}) + 6)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\ln (\frac{5}{x - 3}) + 6) = 3 + \frac{5}{e^{(\ln (\frac{5}{x - 3}) + 6) - 6}}$

Step 5.3.3

Combine the opposite terms in $3 + \frac{5}{e^{(\ln (\frac{5}{x - 3}) + 6) - 6}}$ .

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Step 5.3.3.1

Subtract $6$ from $6$ .

$f (\ln (\frac{5}{x - 3}) + 6) = 3 + \frac{5}{e^{\ln (\frac{5}{x - 3}) + 0}}$

Step 5.3.3.2

Add $\ln (\frac{5}{x - 3})$ and $0$ .

$f (\ln (\frac{5}{x - 3}) + 6) = 3 + \frac{5}{e^{\ln (\frac{5}{x - 3})}}$

Step 5.3.4

Simplify each term.

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Step 5.3.4.1

Exponentiation and log are inverse functions.

$f (\ln (\frac{5}{x - 3}) + 6) = 3 + \frac{5}{\frac{5}{x - 3}}$

Step 5.3.4.2

Multiply the numerator by the reciprocal of the denominator.

$f (\ln (\frac{5}{x - 3}) + 6) = 3 + 5 (\frac{x - 3}{5})$

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.

$f (\ln (\frac{5}{x - 3}) + 6) = 3 + 5 (\frac{x - 3}{5})$

Step 5.3.4.3.2

Rewrite the expression.

$f (\ln (\frac{5}{x - 3}) + 6) = 3 + x - 3$

Step 5.3.5

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

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Step 5.3.5.1

Subtract $3$ from $3$ .

$f (\ln (\frac{5}{x - 3}) + 6) = x + 0$

Step 5.3.5.2

Add $x$ and $0$ .

$f (\ln (\frac{5}{x - 3}) + 6) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \ln (\frac{5}{x - 3}) + 6$ is the inverse of $f (x) = 3 + \frac{5}{e^{x - 6}}$ .

$f^{- 1} (x) = \ln (\frac{5}{x - 3}) + 6$